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Properties

Label 4-50e2-1.1-c25e2-0-5
Degree $4$
Conductor $2500$
Sign $1$
Analytic cond. $39203.3$
Root an. cond. $14.0711$
Motivic weight $25$
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  − 8.19e3·2-s − 3.79e5·3-s + 5.03e7·4-s + 3.11e9·6-s + 3.76e8·7-s − 2.74e11·8-s + 8.72e11·9-s + 8.32e12·11-s − 1.91e13·12-s + 1.06e14·13-s − 3.08e12·14-s + 1.40e15·16-s − 1.32e15·17-s − 7.14e15·18-s − 4.77e14·19-s − 1.43e14·21-s − 6.81e16·22-s + 1.15e17·23-s + 1.04e17·24-s − 8.72e17·26-s − 9.29e17·27-s + 1.89e16·28-s + 1.72e18·29-s − 8.68e18·31-s − 6.91e18·32-s − 3.16e18·33-s + 1.08e19·34-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.412·3-s + 3/2·4-s + 0.583·6-s + 0.0102·7-s − 1.41·8-s + 1.02·9-s + 0.799·11-s − 0.618·12-s + 1.26·13-s − 0.0145·14-s + 5/4·16-s − 0.552·17-s − 1.45·18-s − 0.0494·19-s − 0.00424·21-s − 1.13·22-s + 1.09·23-s + 0.583·24-s − 1.79·26-s − 1.19·27-s + 0.0154·28-s + 0.905·29-s − 1.98·31-s − 1.06·32-s − 0.329·33-s + 0.781·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s+25/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2500\)    =    \(2^{2} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(39203.3\)
Root analytic conductor: \(14.0711\)
Motivic weight: \(25\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 2500,\ (\ :25/2, 25/2),\ 1)\)

Particular Values

\(L(13)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{27}{2})\) 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$C_1$ \( ( 1 + p^{12} T )^{2} \)
5 \( 1 \)
good3$D_{4}$ \( 1 + 126616 p T - 332800574 p^{7} T^{2} + 126616 p^{26} T^{3} + p^{50} T^{4} \)
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

−10.62551121825836687047960511456, −9.902187978979255746994516159569, −9.472231901410124564038491773144, −9.039714686875459904781508068143, −8.471425449580224367202345779404, −7.956256082449269039531899838696, −7.20873470411525535635754624127, −6.88846942323541698403999760851, −6.33209612018170452321617318316, −5.87675457435075388923794713264, −5.08725926764375289198714881171, −4.33939269053761495112626053445, −3.77060966191111058636399324325, −3.19169733405070615504727499730, −2.44808228372796036928895478094, −1.62884763983763429371875651334, −1.34046685561010609370492638404, −1.07294077704102346513981733557, 0, 0, 1.07294077704102346513981733557, 1.34046685561010609370492638404, 1.62884763983763429371875651334, 2.44808228372796036928895478094, 3.19169733405070615504727499730, 3.77060966191111058636399324325, 4.33939269053761495112626053445, 5.08725926764375289198714881171, 5.87675457435075388923794713264, 6.33209612018170452321617318316, 6.88846942323541698403999760851, 7.20873470411525535635754624127, 7.956256082449269039531899838696, 8.471425449580224367202345779404, 9.039714686875459904781508068143, 9.472231901410124564038491773144, 9.902187978979255746994516159569, 10.62551121825836687047960511456

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