Properties

Label 2-5e2-5.2-c32-0-27
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.23e4 − 3.23e4i)2-s + (−1.63e7 + 1.63e7i)3-s − 2.19e9i·4-s + 1.06e12·6-s + (2.79e13 + 2.79e13i)7-s + (−2.10e14 + 2.10e14i)8-s + 1.31e15i·9-s + 6.58e16·11-s + (3.59e16 + 3.59e16i)12-s + (2.99e16 − 2.99e16i)13-s − 1.81e18i·14-s + 4.19e18·16-s + (1.92e19 + 1.92e19i)17-s + (4.26e19 − 4.26e19i)18-s − 5.51e20i·19-s + ⋯
L(s)  = 1  + (−0.494 − 0.494i)2-s + (−0.380 + 0.380i)3-s − 0.511i·4-s + 0.375·6-s + (0.842 + 0.842i)7-s + (−0.747 + 0.747i)8-s + 0.710i·9-s + 1.43·11-s + (0.194 + 0.194i)12-s + (0.0449 − 0.0449i)13-s − 0.832i·14-s + 0.227·16-s + (0.396 + 0.396i)17-s + (0.351 − 0.351i)18-s − 1.91i·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

\(L(\frac{33}{2})\) \(\approx\) \(1.789625868\)
\(L(\frac12)\) \(\approx\) \(1.789625868\)
\(L(17)\) 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 \( 1 + (3.23e4 + 3.23e4i)T + 4.29e9iT^{2} \)
3 \( 1 + (1.63e7 - 1.63e7i)T - 1.85e15iT^{2} \)
7 \( 1 + (-2.79e13 - 2.79e13i)T + 1.10e27iT^{2} \)
11 \( 1 - 6.58e16T + 2.11e33T^{2} \)
13 \( 1 + (-2.99e16 + 2.99e16i)T - 4.42e35iT^{2} \)
17 \( 1 + (-1.92e19 - 1.92e19i)T + 2.36e39iT^{2} \)
19 \( 1 + 5.51e20iT - 8.31e40T^{2} \)
23 \( 1 + (3.28e21 - 3.28e21i)T - 3.76e43iT^{2} \)
29 \( 1 + 3.10e23iT - 6.26e46T^{2} \)
31 \( 1 - 1.32e24T + 5.29e47T^{2} \)
37 \( 1 + (2.81e24 + 2.81e24i)T + 1.52e50iT^{2} \)
41 \( 1 - 3.62e25T + 4.06e51T^{2} \)
43 \( 1 + (1.14e26 - 1.14e26i)T - 1.86e52iT^{2} \)
47 \( 1 + (-7.62e24 - 7.62e24i)T + 3.21e53iT^{2} \)
53 \( 1 + (-4.02e27 + 4.02e27i)T - 1.50e55iT^{2} \)
59 \( 1 + 3.60e27iT - 4.64e56T^{2} \)
61 \( 1 + 1.10e28T + 1.35e57T^{2} \)
67 \( 1 + (5.94e27 + 5.94e27i)T + 2.71e58iT^{2} \)
71 \( 1 + 2.05e29T + 1.73e59T^{2} \)
73 \( 1 + (4.18e29 - 4.18e29i)T - 4.22e59iT^{2} \)
79 \( 1 - 2.78e30iT - 5.29e60T^{2} \)
83 \( 1 + (-6.45e30 + 6.45e30i)T - 2.57e61iT^{2} \)
89 \( 1 - 7.48e30iT - 2.40e62T^{2} \)
97 \( 1 + (-1.14e31 - 1.14e31i)T + 3.77e63iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.47779722867812816167789915907, −10.16139735690499284457745772343, −9.167869884802789166348452486438, −8.179613259193056242193302883121, −6.38295918736031323297739503123, −5.33269451488268846852185131296, −4.40143155773005676584590686374, −2.57675161738256267613497212514, −1.67764653993821476903949274813, −0.63286665437814304592948969780, 0.75930888683816011348042916151, 1.46919246604610602969963454046, 3.43006731582348233127001329029, 4.26363012243596300371614790282, 6.07131859606702205934122909030, 6.92156755954666616997828759510, 7.891011523387429821577179730243, 8.934419513033449851600923655754, 10.21254982808092516873583423771, 11.84243847860939412615750129560

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