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Properties

Label 2-5e4-25.11-c1-0-32
Degree $2$
Conductor $625$
Sign $-0.855 + 0.518i$
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  + (1.62 − 1.18i)2-s + (0.934 − 2.87i)3-s + (0.634 − 1.95i)4-s + (−1.88 − 5.79i)6-s + 0.369·7-s + (−0.0341 − 0.105i)8-s + (−4.97 − 3.61i)9-s + (1.41 − 1.02i)11-s + (−5.02 − 3.65i)12-s + (−0.903 − 0.656i)13-s + (0.602 − 0.437i)14-s + (3.14 + 2.28i)16-s + (1.69 + 5.21i)17-s − 12.3·18-s + (−1.15 − 3.56i)19-s + ⋯
L(s)  = 1  + (1.15 − 0.836i)2-s + (0.539 − 1.66i)3-s + (0.317 − 0.977i)4-s + (−0.768 − 2.36i)6-s + 0.139·7-s + (−0.0120 − 0.0371i)8-s + (−1.65 − 1.20i)9-s + (0.425 − 0.309i)11-s + (−1.45 − 1.05i)12-s + (−0.250 − 0.181i)13-s + (0.161 − 0.117i)14-s + (0.786 + 0.571i)16-s + (0.411 + 1.26i)17-s − 2.92·18-s + (−0.266 − 0.818i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.855 + 0.518i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.855 + 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(625\)    =    \(5^{4}\)
Sign: $-0.855 + 0.518i$
Analytic conductor: \(4.99065\)
Root analytic conductor: \(2.23397\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{625} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 625,\ (\ :1/2),\ -0.855 + 0.518i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.848443 - 3.03878i\)
\(L(\frac12)\) \(\approx\) \(0.848443 - 3.03878i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.62 + 1.18i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (-0.934 + 2.87i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 - 0.369T + 7T^{2} \)
11 \( 1 + (-1.41 + 1.02i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (0.903 + 0.656i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-1.69 - 5.21i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (1.15 + 3.56i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (5.86 - 4.25i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-1.29 + 3.98i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-0.0944 - 0.290i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-7.45 - 5.41i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-3.38 - 2.46i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 7.17T + 43T^{2} \)
47 \( 1 + (-0.250 + 0.770i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (1.20 - 3.72i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-1.50 - 1.09i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (7.83 - 5.69i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (3.85 + 11.8i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-3.60 + 11.1i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (2.77 - 2.01i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-1.75 + 5.41i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-2.20 - 6.78i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (1.23 - 0.899i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-1.97 + 6.08i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.65502174244173550298086778344, −9.354366843189932844490298951050, −8.139856322921205604736350885036, −7.75588092650111566552176533054, −6.36591552787991297318906998864, −5.85040998808774106861504172094, −4.37064332292726702260955836152, −3.26448256692098289375181205722, −2.30790853709880844568292920589, −1.32862344565611094907797622961, 2.72289989585241414377775384998, 3.88644091977787436783972246881, 4.41562321521344626126866624911, 5.25032690603327648451638778054, 6.13081110566838795854645228944, 7.35552557151530478447522271963, 8.284228201993119627020549122625, 9.408768274482266683048643567638, 9.911312665045442230936792055655, 10.84550996819043903228384892345

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