↑

Properties

Label 2-5e2-5.2-c32-0-16
Degree $2$
Conductor $25$
Sign $0.850 + 0.525i$
Analytic cond. $162.166$
Root an. cond. $12.7344$
Motivic weight $32$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−6.54e4 − 6.54e4i)2-s + (1.37e7 − 1.37e7i)3-s + 4.26e9i·4-s − 1.80e12·6-s + (2.41e12 + 2.41e12i)7-s + (−2.14e12 + 2.14e12i)8-s + 1.47e15i·9-s + 4.03e16·11-s + (5.86e16 + 5.86e16i)12-s + (2.33e17 − 2.33e17i)13-s − 3.15e17i·14-s + 1.85e19·16-s + (−5.44e19 − 5.44e19i)17-s + (9.64e19 − 9.64e19i)18-s + 2.36e20i·19-s + ⋯
L(s)  = 1  + (−0.998 − 0.998i)2-s + (0.319 − 0.319i)3-s + 0.992i·4-s − 0.638·6-s + (0.0725 + 0.0725i)7-s + (−0.00763 + 0.00763i)8-s + 0.795i·9-s + 0.878·11-s + (0.317 + 0.317i)12-s + (0.351 − 0.351i)13-s − 0.144i·14-s + 1.00·16-s + (−1.11 − 1.11i)17-s + (0.793 − 0.793i)18-s + 0.819i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $0.850 + 0.525i$
Analytic conductor: \(162.166\)
Root analytic conductor: \(12.7344\)
Motivic weight: \(32\)
Rational: no
Arithmetic: yes
Character: $\chi_{25} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 25,\ (\ :16),\ 0.850 + 0.525i)\)

Particular Values

\(L(\frac{33}{2})\) \(\approx\) \(1.179883032\)
\(L(\frac12)\) \(\approx\) \(1.179883032\)
\(L(17)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
good2 \( 1 + (6.54e4 + 6.54e4i)T + 4.29e9iT^{2} \)
3 \( 1 + (-1.37e7 + 1.37e7i)T - 1.85e15iT^{2} \)
7 \( 1 + (-2.41e12 - 2.41e12i)T + 1.10e27iT^{2} \)
11 \( 1 - 4.03e16T + 2.11e33T^{2} \)
13 \( 1 + (-2.33e17 + 2.33e17i)T - 4.42e35iT^{2} \)
17 \( 1 + (5.44e19 + 5.44e19i)T + 2.36e39iT^{2} \)
19 \( 1 - 2.36e20iT - 8.31e40T^{2} \)
23 \( 1 + (9.56e20 - 9.56e20i)T - 3.76e43iT^{2} \)
29 \( 1 - 1.73e22iT - 6.26e46T^{2} \)
31 \( 1 - 2.11e23T + 5.29e47T^{2} \)
37 \( 1 + (1.04e25 + 1.04e25i)T + 1.52e50iT^{2} \)
41 \( 1 + 3.86e25T + 4.06e51T^{2} \)
43 \( 1 + (-1.15e26 + 1.15e26i)T - 1.86e52iT^{2} \)
47 \( 1 + (5.56e26 + 5.56e26i)T + 3.21e53iT^{2} \)
53 \( 1 + (1.97e27 - 1.97e27i)T - 1.50e55iT^{2} \)
59 \( 1 - 2.24e28iT - 4.64e56T^{2} \)
61 \( 1 - 4.54e28T + 1.35e57T^{2} \)
67 \( 1 + (-1.71e29 - 1.71e29i)T + 2.71e58iT^{2} \)
71 \( 1 - 4.36e29T + 1.73e59T^{2} \)
73 \( 1 + (6.60e29 - 6.60e29i)T - 4.22e59iT^{2} \)
79 \( 1 + 2.22e30iT - 5.29e60T^{2} \)
83 \( 1 + (1.54e30 - 1.54e30i)T - 2.57e61iT^{2} \)
89 \( 1 - 2.18e31iT - 2.40e62T^{2} \)
97 \( 1 + (-7.63e31 - 7.63e31i)T + 3.77e63iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.17052681395773883057497779950, −10.16144462623789192250921565955, −9.014491595998014327498926087851, −8.240478500646777911420556486609, −6.97292422912369349916925535501, −5.36226641799800430435414964766, −3.75058862278447817796893009727, −2.49248061840592079929673646267, −1.75337765639739741626149841229, −0.72120217224917010817749984445, 0.43450160052091724461219600063, 1.57303627110420325034778934237, 3.31638784898051812359695634424, 4.41828824994830936955686731768, 6.26915091693994495985853244468, 6.74355645216988453951299019980, 8.255173769756188334258660698716, 8.978775377576783739937005138807, 9.800045616865819201961836563184, 11.23793784942443469481089415186

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