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Properties

Label 2-5e4-25.16-c1-0-24
Degree $2$
Conductor $625$
Sign $0.756 - 0.653i$
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  + (1.36 + 0.989i)2-s + (0.219 + 0.675i)3-s + (0.257 + 0.793i)4-s + (−0.369 + 1.13i)6-s + 4.59·7-s + (0.606 − 1.86i)8-s + (2.01 − 1.46i)9-s + (−3.16 − 2.30i)11-s + (−0.479 + 0.348i)12-s + (−0.463 + 0.336i)13-s + (6.25 + 4.54i)14-s + (4.02 − 2.92i)16-s + (0.0718 − 0.221i)17-s + 4.20·18-s + (−1.71 + 5.27i)19-s + ⋯
L(s)  = 1  + (0.963 + 0.699i)2-s + (0.126 + 0.390i)3-s + (0.128 + 0.396i)4-s + (−0.150 + 0.464i)6-s + 1.73·7-s + (0.214 − 0.660i)8-s + (0.672 − 0.488i)9-s + (−0.954 − 0.693i)11-s + (−0.138 + 0.100i)12-s + (−0.128 + 0.0934i)13-s + (1.67 + 1.21i)14-s + (1.00 − 0.730i)16-s + (0.0174 − 0.0536i)17-s + 0.990·18-s + (−0.393 + 1.21i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 - 0.653i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.756 - 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(625\)    =    \(5^{4}\)
Sign: $0.756 - 0.653i$
Analytic conductor: \(4.99065\)
Root analytic conductor: \(2.23397\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{625} (376, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 625,\ (\ :1/2),\ 0.756 - 0.653i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.72792 + 1.01450i\)
\(L(\frac12)\) \(\approx\) \(2.72792 + 1.01450i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.36 - 0.989i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (-0.219 - 0.675i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 - 4.59T + 7T^{2} \)
11 \( 1 + (3.16 + 2.30i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (0.463 - 0.336i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.0718 + 0.221i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (1.71 - 5.27i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (3.99 + 2.89i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-1.27 - 3.92i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (1.08 - 3.32i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (4.38 - 3.18i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (8.43 - 6.12i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 1.38T + 43T^{2} \)
47 \( 1 + (0.284 + 0.875i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (0.380 + 1.17i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (3.64 - 2.64i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-9.43 - 6.85i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-0.912 + 2.80i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (-0.990 - 3.04i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (8.32 + 6.04i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (2.97 + 9.14i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-3.23 + 9.97i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (5.87 + 4.26i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-2.57 - 7.91i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.49341621705367895400853365594, −10.16535423680581038780363621477, −8.654864170290307572651011715003, −8.000160109145109279538668226608, −7.04910767380269976945556306757, −5.95962340207987981448833001889, −5.05185393811759076647107712960, −4.47751164750504333947314399274, −3.42082576255539252772223347361, −1.55650435290674209363416446428, 1.81884238139291450276606575765, 2.39480150911944505104152145064, 4.07014389780750593539387652897, 4.83856167957357399173822746559, 5.40856220957568962513958784852, 7.13504323718575792858559227915, 7.88158008064251322817371520225, 8.461892839730028211819440357890, 10.00934231152284684996705415651, 10.82169479111285178274958614250

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