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Properties

Label 2-5e4-25.21-c1-0-24
Degree $2$
Conductor $625$
Sign $0.998 + 0.0627i$
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  + (0.350 − 1.07i)2-s + (2.10 + 1.52i)3-s + (0.578 + 0.420i)4-s + (2.38 − 1.73i)6-s + 0.407·7-s + (2.48 − 1.80i)8-s + (1.16 + 3.58i)9-s + (0.618 − 1.90i)11-s + (0.574 + 1.76i)12-s + (−0.216 − 0.666i)13-s + (0.142 − 0.439i)14-s + (−0.636 − 1.95i)16-s + (−1.28 + 0.930i)17-s + 4.27·18-s + (−4.00 + 2.90i)19-s + ⋯
L(s)  = 1  + (0.247 − 0.762i)2-s + (1.21 + 0.883i)3-s + (0.289 + 0.210i)4-s + (0.974 − 0.708i)6-s + 0.153·7-s + (0.880 − 0.639i)8-s + (0.388 + 1.19i)9-s + (0.186 − 0.573i)11-s + (0.165 + 0.510i)12-s + (−0.0600 − 0.184i)13-s + (0.0381 − 0.117i)14-s + (−0.159 − 0.489i)16-s + (−0.310 + 0.225i)17-s + 1.00·18-s + (−0.918 + 0.667i)19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(625\)    =    \(5^{4}\)
Sign: $0.998 + 0.0627i$
Analytic conductor: \(4.99065\)
Root analytic conductor: \(2.23397\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{625} (501, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 625,\ (\ :1/2),\ 0.998 + 0.0627i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.85445 - 0.0897048i\)
\(L(\frac12)\) \(\approx\) \(2.85445 - 0.0897048i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.350 + 1.07i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (-2.10 - 1.52i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 - 0.407T + 7T^{2} \)
11 \( 1 + (-0.618 + 1.90i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (0.216 + 0.666i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (1.28 - 0.930i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (4.00 - 2.90i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-0.371 + 1.14i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (4.45 + 3.23i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (6.63 - 4.82i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.58 + 4.88i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-2.22 - 6.86i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 9.16T + 43T^{2} \)
47 \( 1 + (-1.03 - 0.748i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-4.10 - 2.98i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (2.00 + 6.18i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (2.91 - 8.95i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-2.49 + 1.81i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (5.55 + 4.03i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.168 + 0.518i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (4.43 + 3.22i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (0.788 - 0.572i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (0.700 - 2.15i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (12.3 + 8.94i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.70159988720440540838537301188, −9.810138098017687986965024520949, −8.953735416150162779043136036348, −8.185297535227790430555932804504, −7.31521113177710688759025636005, −5.97498899787243804315826596554, −4.45192882199721753356092978150, −3.78014178041041328541001984294, −2.92691648696213833964312043828, −1.88820072757179948285351013190, 1.70821685253798856210622383308, 2.51108139193830493086406641178, 4.04237674213202806786269150983, 5.25809492611974326162121386764, 6.44283441826976399027016511917, 7.19835826285656121968363641741, 7.67380731053190388000075425431, 8.693429872531544992928796427021, 9.380697666433318060785247071445, 10.65555233163033976341900286802

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