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Properties

Label 2-5e4-25.14-c1-0-14
Degree $2$
Conductor $625$
Sign $0.904 - 0.425i$
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  + (0.951 + 1.30i)2-s + (−0.951 − 0.309i)3-s + (−0.190 + 0.587i)4-s + (−0.499 − 1.53i)6-s + 0.618i·7-s + (2.12 − 0.690i)8-s + (−1.61 − 1.17i)9-s + (4.23 − 3.07i)11-s + (0.363 − 0.5i)12-s + (−1.08 + 1.5i)13-s + (−0.809 + 0.587i)14-s + (3.92 + 2.85i)16-s + (4.97 − 1.61i)17-s − 3.23i·18-s + (−0.263 − 0.812i)19-s + ⋯
L(s)  = 1  + (0.672 + 0.925i)2-s + (−0.549 − 0.178i)3-s + (−0.0954 + 0.293i)4-s + (−0.204 − 0.628i)6-s + 0.233i·7-s + (0.751 − 0.244i)8-s + (−0.539 − 0.391i)9-s + (1.27 − 0.927i)11-s + (0.104 − 0.144i)12-s + (−0.302 + 0.416i)13-s + (−0.216 + 0.157i)14-s + (0.981 + 0.713i)16-s + (1.20 − 0.392i)17-s − 0.762i·18-s + (−0.0605 − 0.186i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 - 0.425i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.904 - 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(625\)    =    \(5^{4}\)
Sign: $0.904 - 0.425i$
Analytic conductor: \(4.99065\)
Root analytic conductor: \(2.23397\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{625} (374, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 625,\ (\ :1/2),\ 0.904 - 0.425i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.90425 + 0.425650i\)
\(L(\frac12)\) \(\approx\) \(1.90425 + 0.425650i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.951 - 1.30i)T + (-0.618 + 1.90i)T^{2} \)
3 \( 1 + (0.951 + 0.309i)T + (2.42 + 1.76i)T^{2} \)
7 \( 1 - 0.618iT - 7T^{2} \)
11 \( 1 + (-4.23 + 3.07i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (1.08 - 1.5i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-4.97 + 1.61i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (0.263 + 0.812i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-2.21 - 3.04i)T + (-7.10 + 21.8i)T^{2} \)
29 \( 1 + (-1.11 + 3.44i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (0.927 + 2.85i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (0.138 - 0.190i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (-0.618 - 0.449i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 4.85iT - 43T^{2} \)
47 \( 1 + (-0.587 - 0.190i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-3.30 - 1.07i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (8.78 + 6.37i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (7.04 - 5.11i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (4.53 - 1.47i)T + (54.2 - 39.3i)T^{2} \)
71 \( 1 + (2.04 - 6.29i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (5.29 + 7.28i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-2.5 + 7.69i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (5.93 - 1.92i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + (7.23 - 5.25i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (3.66 + 1.19i)T + (78.4 + 57.0i)T^{2} \)
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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

−10.89201664293345902662162744162, −9.667260012776954460607931732183, −8.868707873013650607607302508857, −7.71202474113708882806347228273, −6.81364930079279416701321979016, −6.01867105367357002837927865213, −5.54789673238032596467576588877, −4.36832528999113139985474841193, −3.22688244565535515838487044744, −1.14873653393707835910480627989, 1.44348615768788869020512657310, 2.82986446107966829873626040067, 3.92272647336488407525183372518, 4.79727146022757316118006356246, 5.68317037312813902515730206374, 6.92516502666575086214999183647, 7.83867529397518691573048229207, 8.972856175512381322614990674212, 10.19693118630847661025805241170, 10.59723001573234191559833191710

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