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Properties

Label 2-5e4-25.6-c1-0-24
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  + (−0.190 − 0.587i)2-s + (0.809 − 0.587i)3-s + (1.30 − 0.951i)4-s + (−0.5 − 0.363i)6-s + 1.61·7-s + (−1.80 − 1.31i)8-s + (−0.618 + 1.90i)9-s + (−0.236 − 0.726i)11-s + (0.5 − 1.53i)12-s + (1.5 − 4.61i)13-s + (−0.309 − 0.951i)14-s + (0.572 − 1.76i)16-s + (−0.618 − 0.449i)17-s + 1.23·18-s + (4.73 + 3.44i)19-s + ⋯
L(s)  = 1  + (−0.135 − 0.415i)2-s + (0.467 − 0.339i)3-s + (0.654 − 0.475i)4-s + (−0.204 − 0.148i)6-s + 0.611·7-s + (−0.639 − 0.464i)8-s + (−0.206 + 0.634i)9-s + (−0.0711 − 0.219i)11-s + (0.144 − 0.444i)12-s + (0.416 − 1.28i)13-s + (−0.0825 − 0.254i)14-s + (0.143 − 0.440i)16-s + (−0.149 − 0.108i)17-s + 0.291·18-s + (1.08 + 0.789i)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: $\chi_{625} (126, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 625,\ (\ :1/2),\ 0.0627 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41518 - 1.32894i\)
\(L(\frac12)\) \(\approx\) \(1.41518 - 1.32894i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.190 + 0.587i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-0.809 + 0.587i)T + (0.927 - 2.85i)T^{2} \)
7 \( 1 - 1.61T + 7T^{2} \)
11 \( 1 + (0.236 + 0.726i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-1.5 + 4.61i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.618 + 0.449i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-4.73 - 3.44i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (2.54 + 7.83i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-1.11 + 0.812i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-2.42 - 1.76i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (1.30 - 4.02i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.61 - 4.97i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 1.85T + 43T^{2} \)
47 \( 1 + (1.30 - 0.951i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-4.42 + 3.21i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.28 - 3.94i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (1.45 + 4.47i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-7.47 - 5.42i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (-3.54 + 2.57i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-2.78 - 8.55i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (2.5 - 1.81i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (1.42 + 1.03i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-2.76 - 8.50i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-2.30 + 1.67i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.46997073586398874018823272540, −9.788091177889474100556298202608, −8.345049474287985583787110884860, −8.057008358157275654689505230412, −6.88400677695964226748609894547, −5.84421863164464804575837053788, −4.96554661938319594675914135405, −3.28452001713052722585969051794, −2.39848056517299257492800654530, −1.16163044505514384670249480912, 1.85002432249993949617277672156, 3.16952633195153945371273272404, 4.10787357752464197156482770503, 5.44284442383642862502591924466, 6.52783574932759479109328076942, 7.30673839466579759375087409742, 8.180207759049115547447863535447, 9.033446109587493182382210016913, 9.636841413913338928168046609609, 10.99836742395445770156238663983

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