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Properties

Label 2-5e4-25.16-c1-0-4
Degree $2$
Conductor $625$
Sign $0.0209 - 0.999i$
Analytic cond. $4.99065$
Root an. cond. $2.23397$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.264 − 0.192i)2-s + (0.530 + 1.63i)3-s + (−0.585 − 1.80i)4-s + (0.173 − 0.533i)6-s − 3.42·7-s + (−0.393 + 1.20i)8-s + (0.0465 − 0.0337i)9-s + (4.32 + 3.13i)11-s + (2.62 − 1.90i)12-s + (−2.84 + 2.06i)13-s + (0.905 + 0.657i)14-s + (−2.72 + 1.98i)16-s + (−0.790 + 2.43i)17-s − 0.0187·18-s + (−0.626 + 1.92i)19-s + ⋯
L(s)  = 1  + (−0.186 − 0.135i)2-s + (0.306 + 0.941i)3-s + (−0.292 − 0.900i)4-s + (0.0707 − 0.217i)6-s − 1.29·7-s + (−0.138 + 0.427i)8-s + (0.0155 − 0.0112i)9-s + (1.30 + 0.946i)11-s + (0.758 − 0.551i)12-s + (−0.790 + 0.574i)13-s + (0.241 + 0.175i)14-s + (−0.681 + 0.495i)16-s + (−0.191 + 0.590i)17-s − 0.00442·18-s + (−0.143 + 0.442i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0209 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0209 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(625\)    =    \(5^{4}\)
Sign: $0.0209 - 0.999i$
Analytic conductor: \(4.99065\)
Root analytic conductor: \(2.23397\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{625} (376, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 625,\ (\ :1/2),\ 0.0209 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.738379 + 0.723074i\)
\(L(\frac12)\) \(\approx\) \(0.738379 + 0.723074i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.264 + 0.192i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (-0.530 - 1.63i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + 3.42T + 7T^{2} \)
11 \( 1 + (-4.32 - 3.13i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (2.84 - 2.06i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (0.790 - 2.43i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (0.626 - 1.92i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-6.12 - 4.45i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-1.46 - 4.51i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-0.501 + 1.54i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.0108 - 0.00789i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (7.82 - 5.68i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 2.32T + 43T^{2} \)
47 \( 1 + (2.14 + 6.60i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (0.532 + 1.63i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-0.0179 + 0.0130i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (3.16 + 2.30i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (1.27 - 3.91i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (-0.722 - 2.22i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-1.22 - 0.889i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-0.131 - 0.405i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-1.86 + 5.74i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (-4.92 - 3.58i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-4.94 - 15.2i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.39234925624525874148537885141, −9.786169020793100221505594307135, −9.411287926120004211573182667505, −8.752426678657261440148932420591, −6.95914472865921463254259972373, −6.48366497890689578017003345605, −5.12164084164083219486372938191, −4.25836244318537856081118466876, −3.30978180823655544114318869982, −1.61551425853272049366990283496, 0.60562834112357745236909079259, 2.66040532712428355907613991819, 3.40763906098779099588759855733, 4.70816782953179489205490750997, 6.39845731461681250575068920451, 6.83956949175318558552402583728, 7.66813416518213575662185220435, 8.689449010981824586376460308307, 9.211524999232309788131812070484, 10.21328755785428974758565357772

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