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Properties

Label 4-2e2-1.1-c33e2-0-0
Degree $4$
Conductor $4$
Sign $1$
Analytic cond. $190.345$
Root an. cond. $3.71437$
Motivic weight $33$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.31e5·2-s + 8.35e6·3-s + 1.28e10·4-s − 5.33e9·5-s + 1.09e12·6-s + 1.32e14·7-s + 1.12e15·8-s − 2.16e15·9-s − 6.98e14·10-s − 1.58e17·11-s + 1.07e17·12-s + 4.98e18·13-s + 1.73e19·14-s − 4.45e16·15-s + 9.22e19·16-s + 3.02e20·17-s − 2.83e20·18-s + 2.23e21·19-s − 6.87e19·20-s + 1.10e21·21-s − 2.07e22·22-s − 7.14e22·23-s + 9.40e21·24-s − 9.06e22·25-s + 6.53e23·26-s + 9.70e21·27-s + 1.71e24·28-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.112·3-s + 3/2·4-s − 0.0156·5-s + 0.158·6-s + 1.50·7-s + 1.41·8-s − 0.389·9-s − 0.0221·10-s − 1.03·11-s + 0.168·12-s + 2.07·13-s + 2.13·14-s − 0.00175·15-s + 5/4·16-s + 1.50·17-s − 0.550·18-s + 1.77·19-s − 0.0234·20-s + 0.169·21-s − 1.46·22-s − 2.42·23-s + 0.158·24-s − 0.778·25-s + 2.93·26-s + 0.0234·27-s + 2.26·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+33/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $1$
Analytic conductor: \(190.345\)
Root analytic conductor: \(3.71437\)
Motivic weight: \(33\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4,\ (\ :33/2, 33/2),\ 1)\)

Particular Values

\(L(17)\) \(\approx\) \(8.805534018\)
\(L(\frac12)\) \(\approx\) \(8.805534018\)
\(L(\frac{35}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( ( 1 - p^{16} T )^{2} \)
good3$D_{4}$ \( 1 - 2785496 p T + 340418053462 p^{8} T^{2} - 2785496 p^{34} T^{3} + p^{66} T^{4} \)
5$D_{4}$ \( 1 + 1066495332 p T + 29020461120983080502 p^{5} T^{2} + 1066495332 p^{34} T^{3} + p^{66} T^{4} \)
7$D_{4}$ \( 1 - 18959870839408 p T + \)\(75\!\cdots\!14\)\( p^{5} T^{2} - 18959870839408 p^{34} T^{3} + p^{66} T^{4} \)
11$D_{4}$ \( 1 + 14373224674793016 p T + \)\(75\!\cdots\!86\)\( p^{2} T^{2} + 14373224674793016 p^{34} T^{3} + p^{66} T^{4} \)
13$D_{4}$ \( 1 - 4983398515499093788 T + \)\(95\!\cdots\!18\)\( p^{2} T^{2} - 4983398515499093788 p^{33} T^{3} + p^{66} T^{4} \)
17$D_{4}$ \( 1 - \)\(30\!\cdots\!76\)\( T + \)\(57\!\cdots\!54\)\( p T^{2} - \)\(30\!\cdots\!76\)\( p^{33} T^{3} + p^{66} T^{4} \)
19$D_{4}$ \( 1 - \)\(22\!\cdots\!00\)\( T + \)\(14\!\cdots\!22\)\( p T^{2} - \)\(22\!\cdots\!00\)\( p^{33} T^{3} + p^{66} T^{4} \)
23$D_{4}$ \( 1 + 5870351094364027056 p^{3} T + \)\(56\!\cdots\!98\)\( p^{2} T^{2} + 5870351094364027056 p^{36} T^{3} + p^{66} T^{4} \)
29$D_{4}$ \( 1 + \)\(87\!\cdots\!80\)\( p T + \)\(31\!\cdots\!58\)\( p^{2} T^{2} + \)\(87\!\cdots\!80\)\( p^{34} T^{3} + p^{66} T^{4} \)
31$D_{4}$ \( 1 + \)\(13\!\cdots\!16\)\( p T + \)\(38\!\cdots\!26\)\( p^{2} T^{2} + \)\(13\!\cdots\!16\)\( p^{34} T^{3} + p^{66} T^{4} \)
37$D_{4}$ \( 1 - \)\(11\!\cdots\!76\)\( T + \)\(10\!\cdots\!38\)\( T^{2} - \)\(11\!\cdots\!76\)\( p^{33} T^{3} + p^{66} T^{4} \)
41$D_{4}$ \( 1 + \)\(16\!\cdots\!76\)\( T + \)\(11\!\cdots\!86\)\( T^{2} + \)\(16\!\cdots\!76\)\( p^{33} T^{3} + p^{66} T^{4} \)
43$D_{4}$ \( 1 + \)\(36\!\cdots\!52\)\( T + \)\(15\!\cdots\!62\)\( T^{2} + \)\(36\!\cdots\!52\)\( p^{33} T^{3} + p^{66} T^{4} \)
47$D_{4}$ \( 1 - \)\(28\!\cdots\!56\)\( T + \)\(29\!\cdots\!38\)\( T^{2} - \)\(28\!\cdots\!56\)\( p^{33} T^{3} + p^{66} T^{4} \)
53$D_{4}$ \( 1 + \)\(43\!\cdots\!92\)\( T + \)\(15\!\cdots\!62\)\( T^{2} + \)\(43\!\cdots\!92\)\( p^{33} T^{3} + p^{66} T^{4} \)
59$D_{4}$ \( 1 - \)\(18\!\cdots\!60\)\( T + \)\(42\!\cdots\!58\)\( T^{2} - \)\(18\!\cdots\!60\)\( p^{33} T^{3} + p^{66} T^{4} \)
61$D_{4}$ \( 1 - \)\(46\!\cdots\!04\)\( T + \)\(21\!\cdots\!66\)\( T^{2} - \)\(46\!\cdots\!04\)\( p^{33} T^{3} + p^{66} T^{4} \)
67$D_{4}$ \( 1 + \)\(74\!\cdots\!64\)\( T + \)\(37\!\cdots\!98\)\( T^{2} + \)\(74\!\cdots\!64\)\( p^{33} T^{3} + p^{66} T^{4} \)
71$D_{4}$ \( 1 - \)\(26\!\cdots\!64\)\( T + \)\(17\!\cdots\!46\)\( T^{2} - \)\(26\!\cdots\!64\)\( p^{33} T^{3} + p^{66} T^{4} \)
73$D_{4}$ \( 1 + \)\(52\!\cdots\!32\)\( T + \)\(68\!\cdots\!22\)\( T^{2} + \)\(52\!\cdots\!32\)\( p^{33} T^{3} + p^{66} T^{4} \)
79$D_{4}$ \( 1 + \)\(26\!\cdots\!80\)\( T + \)\(98\!\cdots\!78\)\( T^{2} + \)\(26\!\cdots\!80\)\( p^{33} T^{3} + p^{66} T^{4} \)
83$D_{4}$ \( 1 + \)\(37\!\cdots\!92\)\( T + \)\(35\!\cdots\!42\)\( T^{2} + \)\(37\!\cdots\!92\)\( p^{33} T^{3} + p^{66} T^{4} \)
89$D_{4}$ \( 1 - \)\(30\!\cdots\!80\)\( T + \)\(63\!\cdots\!38\)\( T^{2} - \)\(30\!\cdots\!80\)\( p^{33} T^{3} + p^{66} T^{4} \)
97$D_{4}$ \( 1 - \)\(68\!\cdots\!16\)\( T + \)\(68\!\cdots\!18\)\( T^{2} - \)\(68\!\cdots\!16\)\( p^{33} T^{3} + p^{66} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.53058432789469914546486500619, −20.31590919979115517195144884960, −18.46052819043271712489008514299, −17.94302614026743016454597574653, −16.24586336183967891244723904353, −15.80122209652185855259865700661, −14.47359830577973256871647887775, −14.02982978564107361760562836626, −13.16845213122273547496469714219, −11.77033596894310440026522816141, −11.35670696479941157197490227224, −10.16076196227441268183542950897, −8.145887549812245160496356076384, −7.68497069964222198226327041299, −5.72735934513992236494353208315, −5.54247256695501639065762808126, −4.11358545594269031435804728242, −3.30274134422974868219869663836, −1.99487032221332408813705573070, −1.10233445261829048797961018763, 1.10233445261829048797961018763, 1.99487032221332408813705573070, 3.30274134422974868219869663836, 4.11358545594269031435804728242, 5.54247256695501639065762808126, 5.72735934513992236494353208315, 7.68497069964222198226327041299, 8.145887549812245160496356076384, 10.16076196227441268183542950897, 11.35670696479941157197490227224, 11.77033596894310440026522816141, 13.16845213122273547496469714219, 14.02982978564107361760562836626, 14.47359830577973256871647887775, 15.80122209652185855259865700661, 16.24586336183967891244723904353, 17.94302614026743016454597574653, 18.46052819043271712489008514299, 20.31590919979115517195144884960, 20.53058432789469914546486500619

Graph of the $Z$-function along the critical line