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Properties

Label 4-2e2-1.1-c25e2-0-0
Degree $4$
Conductor $4$
Sign $1$
Analytic cond. $62.7253$
Root an. cond. $2.81423$
Motivic weight $25$
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  + 8.19e3·2-s + 3.79e5·3-s + 5.03e7·4-s + 7.41e8·5-s + 3.11e9·6-s − 3.76e8·7-s + 2.74e11·8-s + 8.72e11·9-s + 6.07e12·10-s + 8.32e12·11-s + 1.91e13·12-s − 1.06e14·13-s − 3.08e12·14-s + 2.81e14·15-s + 1.40e15·16-s + 1.32e15·17-s + 7.14e15·18-s − 4.77e14·19-s + 3.73e16·20-s − 1.43e14·21-s + 6.81e16·22-s − 1.15e17·23-s + 1.04e17·24-s + 7.49e16·25-s − 8.72e17·26-s + 9.29e17·27-s − 1.89e16·28-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.412·3-s + 3/2·4-s + 1.35·5-s + 0.583·6-s − 0.0102·7-s + 1.41·8-s + 1.02·9-s + 1.92·10-s + 0.799·11-s + 0.618·12-s − 1.26·13-s − 0.0145·14-s + 0.560·15-s + 5/4·16-s + 0.552·17-s + 1.45·18-s − 0.0494·19-s + 2.03·20-s − 0.00424·21-s + 1.13·22-s − 1.09·23-s + 0.583·24-s + 0.251·25-s − 1.79·26-s + 1.19·27-s − 0.0154·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+25/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $1$
Analytic conductor: \(62.7253\)
Root analytic conductor: \(2.81423\)
Motivic weight: \(25\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4,\ (\ :25/2, 25/2),\ 1)\)

Particular Values

\(L(13)\) \(\approx\) \(8.569501264\)
\(L(\frac12)\) \(\approx\) \(8.569501264\)
\(L(\frac{27}{2})\) 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$C_1$ \( ( 1 - p^{12} T )^{2} \)
good3$D_{4}$ \( 1 - 126616 p T - 332800574 p^{7} T^{2} - 126616 p^{26} T^{3} + p^{50} T^{4} \)
5$D_{4}$ \( 1 - 29678124 p^{2} T + 760941305600974 p^{4} T^{2} - 29678124 p^{27} T^{3} + p^{50} T^{4} \)
7$D_{4}$ \( 1 + 53790992 p T + 736567609831341198 p^{4} T^{2} + 53790992 p^{26} T^{3} + p^{50} T^{4} \)
11$D_{4}$ \( 1 - 756639510024 p T + \)\(17\!\cdots\!46\)\( p^{3} T^{2} - 756639510024 p^{26} T^{3} + p^{50} T^{4} \)
13$D_{4}$ \( 1 + 106467053152292 T + \)\(11\!\cdots\!54\)\( p T^{2} + 106467053152292 p^{25} T^{3} + p^{50} T^{4} \)
17$D_{4}$ \( 1 - 1327878920113956 T + \)\(39\!\cdots\!94\)\( p T^{2} - 1327878920113956 p^{25} T^{3} + p^{50} T^{4} \)
19$D_{4}$ \( 1 + 477079242949400 T + \)\(83\!\cdots\!42\)\( p T^{2} + 477079242949400 p^{25} T^{3} + p^{50} T^{4} \)
23$D_{4}$ \( 1 + 5013261990498864 p T + \)\(43\!\cdots\!58\)\( p^{2} T^{2} + 5013261990498864 p^{26} T^{3} + p^{50} T^{4} \)
29$D_{4}$ \( 1 - 1724412645206435580 T + \)\(30\!\cdots\!98\)\( T^{2} - 1724412645206435580 p^{25} T^{3} + p^{50} T^{4} \)
31$D_{4}$ \( 1 + 8688082288351126976 T + \)\(57\!\cdots\!46\)\( T^{2} + 8688082288351126976 p^{25} T^{3} + p^{50} T^{4} \)
37$D_{4}$ \( 1 + 34908364049750170484 T + \)\(27\!\cdots\!78\)\( T^{2} + 34908364049750170484 p^{25} T^{3} + p^{50} T^{4} \)
41$D_{4}$ \( 1 - 83014324355953468884 T + \)\(29\!\cdots\!66\)\( T^{2} - 83014324355953468884 p^{25} T^{3} + p^{50} T^{4} \)
43$D_{4}$ \( 1 - 44539608583471901848 T + \)\(11\!\cdots\!62\)\( T^{2} - 44539608583471901848 p^{25} T^{3} + p^{50} T^{4} \)
47$D_{4}$ \( 1 - 39773632596907970208 p T + \)\(19\!\cdots\!58\)\( T^{2} - 39773632596907970208 p^{26} T^{3} + p^{50} T^{4} \)
53$D_{4}$ \( 1 + \)\(32\!\cdots\!32\)\( T + \)\(28\!\cdots\!42\)\( T^{2} + \)\(32\!\cdots\!32\)\( p^{25} T^{3} + p^{50} T^{4} \)
59$D_{4}$ \( 1 + \)\(17\!\cdots\!40\)\( T + \)\(42\!\cdots\!98\)\( T^{2} + \)\(17\!\cdots\!40\)\( p^{25} T^{3} + p^{50} T^{4} \)
61$D_{4}$ \( 1 - \)\(33\!\cdots\!44\)\( T + \)\(85\!\cdots\!86\)\( T^{2} - \)\(33\!\cdots\!44\)\( p^{25} T^{3} + p^{50} T^{4} \)
67$D_{4}$ \( 1 - \)\(33\!\cdots\!16\)\( T + \)\(43\!\cdots\!78\)\( T^{2} - \)\(33\!\cdots\!16\)\( p^{25} T^{3} + p^{50} T^{4} \)
71$D_{4}$ \( 1 + \)\(27\!\cdots\!56\)\( T + \)\(49\!\cdots\!86\)\( T^{2} + \)\(27\!\cdots\!56\)\( p^{25} T^{3} + p^{50} T^{4} \)
73$D_{4}$ \( 1 - \)\(31\!\cdots\!48\)\( T + \)\(86\!\cdots\!62\)\( T^{2} - \)\(31\!\cdots\!48\)\( p^{25} T^{3} + p^{50} T^{4} \)
79$D_{4}$ \( 1 - \)\(92\!\cdots\!20\)\( T + \)\(73\!\cdots\!98\)\( T^{2} - \)\(92\!\cdots\!20\)\( p^{25} T^{3} + p^{50} T^{4} \)
83$D_{4}$ \( 1 + \)\(45\!\cdots\!52\)\( T - \)\(76\!\cdots\!38\)\( T^{2} + \)\(45\!\cdots\!52\)\( p^{25} T^{3} + p^{50} T^{4} \)
89$D_{4}$ \( 1 + \)\(23\!\cdots\!20\)\( T + \)\(80\!\cdots\!98\)\( T^{2} + \)\(23\!\cdots\!20\)\( p^{25} T^{3} + p^{50} T^{4} \)
97$D_{4}$ \( 1 - \)\(13\!\cdots\!36\)\( T + \)\(12\!\cdots\!38\)\( T^{2} - \)\(13\!\cdots\!36\)\( p^{25} T^{3} + p^{50} T^{4} \)
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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

−22.15134100365496802842364456197, −21.97766147046738826075335824364, −21.19208389753462498356108644001, −20.24180583354712465316532818185, −19.29205137885697460875049351785, −17.86971452131425100101498252999, −16.80071504643086272845410878330, −15.68178921653690092362764057562, −14.31572441593052936437737236132, −14.14472070512823609426422755833, −12.85583118836617094852809924520, −12.15149674897595662697810873644, −10.39165734775282512659645327459, −9.477013863951213799278245625209, −7.38952923442123633934034670144, −6.25928827265899712557255199877, −5.10664108876688748126544343749, −3.86947813471126057571003514132, −2.40178545892333212608839019732, −1.54121893824147299728369205257, 1.54121893824147299728369205257, 2.40178545892333212608839019732, 3.86947813471126057571003514132, 5.10664108876688748126544343749, 6.25928827265899712557255199877, 7.38952923442123633934034670144, 9.477013863951213799278245625209, 10.39165734775282512659645327459, 12.15149674897595662697810873644, 12.85583118836617094852809924520, 14.14472070512823609426422755833, 14.31572441593052936437737236132, 15.68178921653690092362764057562, 16.80071504643086272845410878330, 17.86971452131425100101498252999, 19.29205137885697460875049351785, 20.24180583354712465316532818185, 21.19208389753462498356108644001, 21.97766147046738826075335824364, 22.15134100365496802842364456197

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