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Properties

Label 2-5e4-25.21-c1-0-29
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  + (0.713 − 2.19i)2-s + (−0.384 − 0.279i)3-s + (−2.69 − 1.95i)4-s + (−0.887 + 0.644i)6-s + 3.03·7-s + (−2.48 + 1.80i)8-s + (−0.857 − 2.63i)9-s + (0.618 − 1.90i)11-s + (0.489 + 1.50i)12-s + (−0.441 − 1.35i)13-s + (2.16 − 6.66i)14-s + (0.136 + 0.420i)16-s + (−1.50 + 1.09i)17-s − 6.40·18-s + (−0.730 + 0.530i)19-s + ⋯
L(s)  = 1  + (0.504 − 1.55i)2-s + (−0.221 − 0.161i)3-s + (−1.34 − 0.979i)4-s + (−0.362 + 0.263i)6-s + 1.14·7-s + (−0.880 + 0.639i)8-s + (−0.285 − 0.879i)9-s + (0.186 − 0.573i)11-s + (0.141 + 0.434i)12-s + (−0.122 − 0.376i)13-s + (0.578 − 1.78i)14-s + (0.0341 + 0.105i)16-s + (−0.365 + 0.265i)17-s − 1.51·18-s + (−0.167 + 0.121i)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

\(L(1)\) \(\approx\) \(0.0542522 + 1.72633i\)
\(L(\frac12)\) \(\approx\) \(0.0542522 + 1.72633i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.713 + 2.19i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (0.384 + 0.279i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 - 3.03T + 7T^{2} \)
11 \( 1 + (-0.618 + 1.90i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (0.441 + 1.35i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (1.50 - 1.09i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.730 - 0.530i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (1.02 - 3.16i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (3.20 + 2.32i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-5.21 + 3.78i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.18 - 3.63i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.566 + 1.74i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 3.59T + 43T^{2} \)
47 \( 1 + (-3.88 - 2.82i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-7.68 - 5.58i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-3.28 - 10.1i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-4.41 + 13.5i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-8.64 + 6.28i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (10.0 + 7.32i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.0827 + 0.254i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-6.93 - 5.03i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-10.2 + 7.41i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (1.47 - 4.53i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-8.05 - 5.85i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.48852818352859961403028727203, −9.565631605378175849078779089700, −8.676165370981576898444985453226, −7.67609790692370243823149462098, −6.24282034855173464293143207501, −5.27970497412643483861620288014, −4.25739644710027466523484457350, −3.35442937919871416152118534426, −2.09663870904445059785933479384, −0.874135343327760339012178452843, 2.12574279254970838203139556932, 4.16162729279095452586644187921, 4.86440167008395626952446415298, 5.47484758946122912609692302529, 6.64762661288928521956290108946, 7.37595730713508127699931005924, 8.237736265508512032249519938516, 8.810459403393577780775048017104, 10.16596855199009546027895899124, 11.11280950253123270755600562877

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