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Properties

Label 2-5e4-25.19-c1-0-31
Degree $2$
Conductor $625$
Sign $0.0627 + 0.998i$
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  + (2.19 − 0.713i)2-s + (−0.279 − 0.384i)3-s + (2.69 − 1.95i)4-s + (−0.887 − 0.644i)6-s − 3.03i·7-s + (1.80 − 2.48i)8-s + (0.857 − 2.63i)9-s + (0.618 + 1.90i)11-s + (−1.50 − 0.489i)12-s + (−1.35 − 0.441i)13-s + (−2.16 − 6.66i)14-s + (0.136 − 0.420i)16-s + (−1.09 + 1.50i)17-s − 6.40i·18-s + (0.730 + 0.530i)19-s + ⋯
L(s)  = 1  + (1.55 − 0.504i)2-s + (−0.161 − 0.221i)3-s + (1.34 − 0.979i)4-s + (−0.362 − 0.263i)6-s − 1.14i·7-s + (0.639 − 0.880i)8-s + (0.285 − 0.879i)9-s + (0.186 + 0.573i)11-s + (−0.434 − 0.141i)12-s + (−0.376 − 0.122i)13-s + (−0.578 − 1.78i)14-s + (0.0341 − 0.105i)16-s + (−0.265 + 0.365i)17-s − 1.51i·18-s + (0.167 + 0.121i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0627 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0627 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(625\)    =    \(5^{4}\)
Sign: $0.0627 + 0.998i$
Analytic conductor: \(4.99065\)
Root analytic conductor: \(2.23397\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{625} (499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 625,\ (\ :1/2),\ 0.0627 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.37750 - 2.23262i\)
\(L(\frac12)\) \(\approx\) \(2.37750 - 2.23262i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.19 + 0.713i)T + (1.61 - 1.17i)T^{2} \)
3 \( 1 + (0.279 + 0.384i)T + (-0.927 + 2.85i)T^{2} \)
7 \( 1 + 3.03iT - 7T^{2} \)
11 \( 1 + (-0.618 - 1.90i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (1.35 + 0.441i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (1.09 - 1.50i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.730 - 0.530i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-3.16 + 1.02i)T + (18.6 - 13.5i)T^{2} \)
29 \( 1 + (-3.20 + 2.32i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-5.21 - 3.78i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (3.63 + 1.18i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.566 - 1.74i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 3.59iT - 43T^{2} \)
47 \( 1 + (2.82 + 3.88i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-5.58 - 7.68i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (3.28 - 10.1i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-4.41 - 13.5i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-6.28 + 8.64i)T + (-20.7 - 63.7i)T^{2} \)
71 \( 1 + (10.0 - 7.32i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (0.254 - 0.0827i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (6.93 - 5.03i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (7.41 - 10.2i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + (-1.47 - 4.53i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (5.85 + 8.05i)T + (-29.9 + 92.2i)T^{2} \)
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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

−10.54426847855673852209654861050, −10.00484837933221991931896569893, −8.677938449328064253023065523248, −7.19469042172964156838856187909, −6.73030500534958358692129383385, −5.66347886759338129272748312496, −4.50806995417856037073059889501, −3.95756110055778982264517968620, −2.81925597549133803027735284493, −1.27648093192640814676643534957, 2.32528922951762497084568107271, 3.31179702025238202596461018028, 4.65961356074932702076863814770, 5.16877787394688168186832408717, 6.04905199217013636977929219312, 6.91148676622059168209771162885, 7.957263011694601796362064033010, 8.979149400128669015225933102439, 10.03448788531217585228168379298, 11.30095685653873714909672022686

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