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Properties

Label 2-5e4-25.16-c1-0-18
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

\(L(1)\) \(\approx\) \(2.87656 + 1.06978i\)
\(L(\frac12)\) \(\approx\) \(2.87656 + 1.06978i\)
\(L(\frac{3}{2})\) 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

−11.37992893536706412567985844080, −9.689637345129575613672850192911, −8.582800678196205561399537586565, −7.30487167778026363053445030933, −7.02106593934452056116736730376, −6.43879050530479902268529798073, −5.17786425771543377542854390390, −4.57870229264439110767268879688, −3.23616427187889281398064180993, −1.64580685529066488018464928618, 1.52471445834741546028738859973, 3.17067521819681901423931212275, 3.95413437065424927235218986405, 4.62346858377466671522299607354, 5.65270002126164466259900983067, 6.21023923669566789139289251050, 7.920883452297659557256991325748, 9.272306934622204708090049735348, 10.03363960103887774253825846985, 10.74978120712202164095960022076

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