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Properties

Label 2-5e4-25.11-c1-0-1
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  + (0.148 − 0.107i)2-s + (−0.454 + 1.39i)3-s + (−0.607 + 1.87i)4-s + (0.0833 + 0.256i)6-s − 3.26·7-s + (0.224 + 0.690i)8-s + (0.674 + 0.489i)9-s + (−1.61 + 1.17i)11-s + (−2.34 − 1.70i)12-s + (0.239 + 0.174i)13-s + (−0.483 + 0.351i)14-s + (−3.07 − 2.23i)16-s + (−1.59 − 4.91i)17-s + 0.152·18-s + (0.534 + 1.64i)19-s + ⋯
L(s)  = 1  + (0.104 − 0.0761i)2-s + (−0.262 + 0.808i)3-s + (−0.303 + 0.935i)4-s + (0.0340 + 0.104i)6-s − 1.23·7-s + (0.0793 + 0.244i)8-s + (0.224 + 0.163i)9-s + (−0.487 + 0.354i)11-s + (−0.675 − 0.491i)12-s + (0.0665 + 0.0483i)13-s + (−0.129 + 0.0938i)14-s + (−0.768 − 0.558i)16-s + (−0.386 − 1.19i)17-s + 0.0359·18-s + (0.122 + 0.377i)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: $\chi_{625} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 625,\ (\ :1/2),\ -0.904 + 0.425i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.103595 - 0.463461i\)
\(L(\frac12)\) \(\approx\) \(0.103595 - 0.463461i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.148 + 0.107i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (0.454 - 1.39i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 + 3.26T + 7T^{2} \)
11 \( 1 + (1.61 - 1.17i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-0.239 - 0.174i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.59 + 4.91i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-0.534 - 1.64i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (0.711 - 0.516i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-1.82 + 5.62i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-1.88 - 5.80i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (6.54 + 4.75i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-0.821 - 0.596i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 3.24T + 43T^{2} \)
47 \( 1 + (-1.30 + 4.01i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (2.50 - 7.70i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (4.80 + 3.48i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (0.740 - 0.538i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-2.12 - 6.55i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (1.84 - 5.67i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (7.14 - 5.19i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (2.39 - 7.38i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-4.48 - 13.8i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (6.08 - 4.42i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (2.07 - 6.39i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.04383095119051804040265730676, −10.05467683582807542353620638071, −9.559194524379681567579814063267, −8.639718826916471985913184351634, −7.51189386641160859246334040002, −6.77359939344280756447750777841, −5.40532754456736821177181456974, −4.50069349865512956302245020122, −3.60842975345267817801450237214, −2.60970622539301118039507228355, 0.25361856198140127864437841069, 1.70981889044240014554306706027, 3.31540511669543602159958471831, 4.60461972244240874854870165524, 5.89768711700079069403823385074, 6.35875046747294065846334799433, 7.14526713685566798578712612110, 8.397876649364017300644094738019, 9.356804459501565880500164725572, 10.14682520090661308623748955134

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