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Properties

Label 2-5e4-25.16-c1-0-5
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  + (−2.00 − 1.45i)2-s + (0.654 + 2.01i)3-s + (1.27 + 3.92i)4-s + (1.61 − 4.98i)6-s − 0.973·7-s + (1.62 − 4.99i)8-s + (−1.19 + 0.870i)9-s + (4.35 + 3.16i)11-s + (−7.05 + 5.12i)12-s + (1.61 − 1.17i)13-s + (1.94 + 1.41i)14-s + (−3.84 + 2.79i)16-s + (−0.631 + 1.94i)17-s + 3.66·18-s + (1.91 − 5.90i)19-s + ⋯
L(s)  = 1  + (−1.41 − 1.02i)2-s + (0.377 + 1.16i)3-s + (0.636 + 1.96i)4-s + (0.660 − 2.03i)6-s − 0.367·7-s + (0.573 − 1.76i)8-s + (−0.399 + 0.290i)9-s + (1.31 + 0.953i)11-s + (−2.03 + 1.48i)12-s + (0.448 − 0.325i)13-s + (0.520 + 0.378i)14-s + (−0.960 + 0.697i)16-s + (−0.153 + 0.471i)17-s + 0.863·18-s + (0.439 − 1.35i)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

\(L(1)\) \(\approx\) \(0.789173 + 0.293490i\)
\(L(\frac12)\) \(\approx\) \(0.789173 + 0.293490i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.00 + 1.45i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (-0.654 - 2.01i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + 0.973T + 7T^{2} \)
11 \( 1 + (-4.35 - 3.16i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-1.61 + 1.17i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (0.631 - 1.94i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-1.91 + 5.90i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (1.56 + 1.13i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-1.48 - 4.58i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (2.05 - 6.31i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-0.791 + 0.575i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (2.21 - 1.60i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 3.99T + 43T^{2} \)
47 \( 1 + (-2.22 - 6.86i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-4.10 - 12.6i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (5.29 - 3.84i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (2.20 + 1.60i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-2.95 + 9.10i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (1.75 + 5.40i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-7.56 - 5.49i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (0.983 + 3.02i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-1.88 + 5.81i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (-2.43 - 1.76i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-1.84 - 5.66i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.56576835867635551982429362287, −9.729996075985700698959285089540, −9.138300076028801382176100064149, −8.773308193728889584515883910916, −7.49531782863898704879890802871, −6.54085703031161288479125857944, −4.72044389855751750495432636579, −3.73074546034950105309062499442, −2.86892688985746358052485169394, −1.36118494580177387872790927087, 0.818837866224618400916646000495, 1.90484035676581351232868059432, 3.72602518078035311257366781349, 5.77729965501022171507395126163, 6.39223288633414606208820228578, 7.07055678309644977649824358691, 7.962514942424040866095181793398, 8.502050032937229348638878924288, 9.370662648346089153543578770348, 10.05531493934162512212926672053

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