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Properties

Label 4-2e4-1.1-c23e2-0-0
Degree $4$
Conductor $16$
Sign $1$
Analytic cond. $179.778$
Root an. cond. $3.66171$
Motivic weight $23$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.70e5·3-s − 9.22e7·5-s + 1.92e8·7-s − 7.53e10·9-s − 1.24e12·11-s − 7.46e12·13-s − 1.57e13·15-s − 3.74e14·17-s − 8.40e14·19-s + 3.27e13·21-s + 6.43e15·23-s + 9.12e15·25-s − 1.45e16·27-s − 6.80e16·29-s − 1.45e17·31-s − 2.12e17·33-s − 1.77e16·35-s + 1.21e18·37-s − 1.27e18·39-s + 5.03e18·41-s + 9.35e17·43-s + 6.94e18·45-s − 8.43e18·47-s − 5.43e19·49-s − 6.39e19·51-s + 3.76e18·53-s + 1.15e20·55-s + ⋯
L(s)  = 1  + 0.555·3-s − 0.845·5-s + 0.0367·7-s − 0.799·9-s − 1.31·11-s − 1.15·13-s − 0.469·15-s − 2.65·17-s − 1.65·19-s + 0.0204·21-s + 1.40·23-s + 0.765·25-s − 0.504·27-s − 1.03·29-s − 1.02·31-s − 0.732·33-s − 0.0310·35-s + 1.11·37-s − 0.641·39-s + 1.42·41-s + 0.153·43-s + 0.675·45-s − 0.497·47-s − 1.98·49-s − 1.47·51-s + 0.0557·53-s + 1.11·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+23/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $1$
Analytic conductor: \(179.778\)
Root analytic conductor: \(3.66171\)
Motivic weight: \(23\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 16,\ (\ :23/2, 23/2),\ 1)\)

Particular Values

\(L(12)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{25}{2})\) 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 \( 1 \)
good3$D_{4}$ \( 1 - 56840 p T + 429554626 p^{5} T^{2} - 56840 p^{24} T^{3} + p^{46} T^{4} \)
5$D_{4}$ \( 1 + 18453204 p T - 985891951106 p^{4} T^{2} + 18453204 p^{24} T^{3} + p^{46} T^{4} \)
7$D_{4}$ \( 1 - 192083440 T + 1109027370312762510 p^{2} T^{2} - 192083440 p^{23} T^{3} + p^{46} T^{4} \)
11$D_{4}$ \( 1 + 113379517800 p T + \)\(11\!\cdots\!22\)\( p^{2} T^{2} + 113379517800 p^{24} T^{3} + p^{46} T^{4} \)
13$D_{4}$ \( 1 + 7460299980980 T + \)\(37\!\cdots\!86\)\( p T^{2} + 7460299980980 p^{23} T^{3} + p^{46} T^{4} \)
17$D_{4}$ \( 1 + 22052785170780 p T + \)\(25\!\cdots\!98\)\( p^{2} T^{2} + 22052785170780 p^{24} T^{3} + p^{46} T^{4} \)
19$D_{4}$ \( 1 + 44229488379608 p T + \)\(36\!\cdots\!26\)\( p T^{2} + 44229488379608 p^{24} T^{3} + p^{46} T^{4} \)
23$D_{4}$ \( 1 - 279711214046640 p T + \)\(40\!\cdots\!90\)\( T^{2} - 279711214046640 p^{24} T^{3} + p^{46} T^{4} \)
29$D_{4}$ \( 1 + 68055499247434452 T + \)\(86\!\cdots\!54\)\( T^{2} + 68055499247434452 p^{23} T^{3} + p^{46} T^{4} \)
31$D_{4}$ \( 1 + 145584514546845248 T + \)\(81\!\cdots\!58\)\( T^{2} + 145584514546845248 p^{23} T^{3} + p^{46} T^{4} \)
37$D_{4}$ \( 1 - 1211894143551157660 T + \)\(25\!\cdots\!30\)\( T^{2} - 1211894143551157660 p^{23} T^{3} + p^{46} T^{4} \)
41$D_{4}$ \( 1 - 5036778367134688692 T + \)\(27\!\cdots\!58\)\( T^{2} - 5036778367134688692 p^{23} T^{3} + p^{46} T^{4} \)
43$D_{4}$ \( 1 - 935180945919935560 T + \)\(58\!\cdots\!14\)\( T^{2} - 935180945919935560 p^{23} T^{3} + p^{46} T^{4} \)
47$D_{4}$ \( 1 + 8431168896277596000 T + \)\(26\!\cdots\!22\)\( T^{2} + 8431168896277596000 p^{23} T^{3} + p^{46} T^{4} \)
53$D_{4}$ \( 1 - 3763137197370204540 T + \)\(89\!\cdots\!30\)\( T^{2} - 3763137197370204540 p^{23} T^{3} + p^{46} T^{4} \)
59$D_{4}$ \( 1 - \)\(34\!\cdots\!16\)\( T + \)\(12\!\cdots\!22\)\( T^{2} - \)\(34\!\cdots\!16\)\( p^{23} T^{3} + p^{46} T^{4} \)
61$D_{4}$ \( 1 + \)\(58\!\cdots\!56\)\( T + \)\(29\!\cdots\!46\)\( T^{2} + \)\(58\!\cdots\!56\)\( p^{23} T^{3} + p^{46} T^{4} \)
67$D_{4}$ \( 1 - \)\(21\!\cdots\!60\)\( T + \)\(16\!\cdots\!70\)\( T^{2} - \)\(21\!\cdots\!60\)\( p^{23} T^{3} + p^{46} T^{4} \)
71$D_{4}$ \( 1 - \)\(38\!\cdots\!96\)\( T + \)\(10\!\cdots\!26\)\( T^{2} - \)\(38\!\cdots\!96\)\( p^{23} T^{3} + p^{46} T^{4} \)
73$D_{4}$ \( 1 + \)\(68\!\cdots\!80\)\( T + \)\(74\!\cdots\!18\)\( T^{2} + \)\(68\!\cdots\!80\)\( p^{23} T^{3} + p^{46} T^{4} \)
79$D_{4}$ \( 1 + \)\(64\!\cdots\!36\)\( T + \)\(63\!\cdots\!02\)\( T^{2} + \)\(64\!\cdots\!36\)\( p^{23} T^{3} + p^{46} T^{4} \)
83$D_{4}$ \( 1 + \)\(18\!\cdots\!00\)\( T + \)\(36\!\cdots\!70\)\( T^{2} + \)\(18\!\cdots\!00\)\( p^{23} T^{3} + p^{46} T^{4} \)
89$D_{4}$ \( 1 - \)\(50\!\cdots\!32\)\( T + \)\(95\!\cdots\!94\)\( T^{2} - \)\(50\!\cdots\!32\)\( p^{23} T^{3} + p^{46} T^{4} \)
97$D_{4}$ \( 1 - \)\(12\!\cdots\!80\)\( T + \)\(12\!\cdots\!10\)\( T^{2} - \)\(12\!\cdots\!80\)\( p^{23} T^{3} + p^{46} T^{4} \)
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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

−18.66823708776659081566104334997, −17.69104253407557431091786668202, −16.90865512170736837885009379172, −15.85125995685105380016151892241, −14.99056386913637168251538220249, −14.65280464776903061269849595007, −13.05390385256653617872416111702, −12.86869627716003820847310222426, −11.09117880818034587591305005118, −11.02153746920171144507848616815, −9.277263723407734261673499419524, −8.527416898533620352503408919712, −7.64835834865326670554535950599, −6.61051309753877980995670795652, −5.06652546589923434490870837084, −4.20900013379266506256658557400, −2.75892311844377540615987774047, −2.25015600057662269581687487080, 0, 0, 2.25015600057662269581687487080, 2.75892311844377540615987774047, 4.20900013379266506256658557400, 5.06652546589923434490870837084, 6.61051309753877980995670795652, 7.64835834865326670554535950599, 8.527416898533620352503408919712, 9.277263723407734261673499419524, 11.02153746920171144507848616815, 11.09117880818034587591305005118, 12.86869627716003820847310222426, 13.05390385256653617872416111702, 14.65280464776903061269849595007, 14.99056386913637168251538220249, 15.85125995685105380016151892241, 16.90865512170736837885009379172, 17.69104253407557431091786668202, 18.66823708776659081566104334997

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