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Properties

Label 2-5e4-25.4-c1-0-5
Degree $2$
Conductor $625$
Sign $0.268 - 0.963i$
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  + (−2.53 − 0.823i)2-s + (0.446 − 0.614i)3-s + (4.12 + 2.99i)4-s + (−1.63 + 1.18i)6-s + 2.04i·7-s + (−4.85 − 6.68i)8-s + (0.748 + 2.30i)9-s + (0.416 − 1.28i)11-s + (3.68 − 1.19i)12-s + (−1.25 + 0.407i)13-s + (1.68 − 5.17i)14-s + (3.65 + 11.2i)16-s + (2.40 + 3.30i)17-s − 6.45i·18-s + (−3.95 + 2.87i)19-s + ⋯
L(s)  = 1  + (−1.79 − 0.582i)2-s + (0.257 − 0.354i)3-s + (2.06 + 1.49i)4-s + (−0.668 + 0.485i)6-s + 0.771i·7-s + (−1.71 − 2.36i)8-s + (0.249 + 0.768i)9-s + (0.125 − 0.386i)11-s + (1.06 − 0.345i)12-s + (−0.348 + 0.113i)13-s + (0.449 − 1.38i)14-s + (0.913 + 2.81i)16-s + (0.582 + 0.801i)17-s − 1.52i·18-s + (−0.906 + 0.658i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.268 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.268 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(625\)    =    \(5^{4}\)
Sign: $0.268 - 0.963i$
Analytic conductor: \(4.99065\)
Root analytic conductor: \(2.23397\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{625} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 625,\ (\ :1/2),\ 0.268 - 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.371197 + 0.281754i\)
\(L(\frac12)\) \(\approx\) \(0.371197 + 0.281754i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.53 + 0.823i)T + (1.61 + 1.17i)T^{2} \)
3 \( 1 + (-0.446 + 0.614i)T + (-0.927 - 2.85i)T^{2} \)
7 \( 1 - 2.04iT - 7T^{2} \)
11 \( 1 + (-0.416 + 1.28i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (1.25 - 0.407i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-2.40 - 3.30i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (3.95 - 2.87i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (2.60 + 0.845i)T + (18.6 + 13.5i)T^{2} \)
29 \( 1 + (3.73 + 2.71i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (5.79 - 4.20i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (8.22 - 2.67i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (3.12 + 9.60i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 2.43iT - 43T^{2} \)
47 \( 1 + (4.45 - 6.12i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (0.365 - 0.502i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-3.50 - 10.7i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-0.200 + 0.615i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-6.42 - 8.84i)T + (-20.7 + 63.7i)T^{2} \)
71 \( 1 + (-2.38 - 1.73i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-12.8 - 4.18i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-1.52 - 1.11i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (1.39 + 1.92i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + (2.26 - 6.97i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (3.40 - 4.69i)T + (-29.9 - 92.2i)T^{2} \)
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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

−10.53517050261767878965103673063, −9.982477009655620227014118340474, −8.871356144367877162397906352021, −8.407268833011762328620479334342, −7.66222674336730247598965742572, −6.76789175473439292643531094532, −5.60910003349340456443869825784, −3.68641612372370129386737463364, −2.36993185944264888191190019151, −1.63463359941913109751784197749, 0.43000789125395950101258789490, 1.95596990633871176983987384646, 3.58564222339384636642213389460, 5.11726581254274346809764498083, 6.47065133810564873700706535038, 7.09276606259270362813331556245, 7.83528824287349697779557733163, 8.828796791680543167873535674319, 9.540192861063317416142319385209, 10.04074211631942305394687180835

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