Properties

Label 2-5e3-25.6-c1-0-2
Degree $2$
Conductor $125$
Sign $0.535 - 0.844i$
Analytic cond. $0.998130$
Root an. cond. $0.999064$
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  + (0.5 + 1.53i)2-s + (0.809 − 0.587i)3-s + (−0.5 + 0.363i)4-s + (1.30 + 0.951i)6-s − 0.618·7-s + (1.80 + 1.31i)8-s + (−0.618 + 1.90i)9-s + (−1.61 − 4.97i)11-s + (−0.190 + 0.587i)12-s + (−0.572 + 1.76i)13-s + (−0.309 − 0.951i)14-s + (−1.50 + 4.61i)16-s + (−4.23 − 3.07i)17-s − 3.23·18-s + (−0.690 − 0.502i)19-s + ⋯
L(s)  = 1  + (0.353 + 1.08i)2-s + (0.467 − 0.339i)3-s + (−0.250 + 0.181i)4-s + (0.534 + 0.388i)6-s − 0.233·7-s + (0.639 + 0.464i)8-s + (−0.206 + 0.634i)9-s + (−0.487 − 1.50i)11-s + (−0.0551 + 0.169i)12-s + (−0.158 + 0.489i)13-s + (−0.0825 − 0.254i)14-s + (−0.375 + 1.15i)16-s + (−1.02 − 0.746i)17-s − 0.762·18-s + (−0.158 − 0.115i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.535 - 0.844i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(125\)    =    \(5^{3}\)
Sign: $0.535 - 0.844i$
Analytic conductor: \(0.998130\)
Root analytic conductor: \(0.999064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{125} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 125,\ (\ :1/2),\ 0.535 - 0.844i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26339 + 0.694559i\)
\(L(\frac12)\) \(\approx\) \(1.26339 + 0.694559i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 1.53i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-0.809 + 0.587i)T + (0.927 - 2.85i)T^{2} \)
7 \( 1 + 0.618T + 7T^{2} \)
11 \( 1 + (1.61 + 4.97i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (0.572 - 1.76i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (4.23 + 3.07i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.690 + 0.502i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (1.16 + 3.57i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-2.92 + 2.12i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-2.42 - 1.76i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.0729 + 0.224i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.236 - 0.726i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 4.85T + 43T^{2} \)
47 \( 1 + (-0.5 + 0.363i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (2.80 - 2.04i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (3.35 - 10.3i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-2.69 - 8.28i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-3.85 - 2.80i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (-5.35 + 3.88i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-2.78 - 8.55i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-6.54 + 4.75i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (5.04 + 3.66i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (2.76 + 8.50i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (3.11 - 2.26i)T + (29.9 - 92.2i)T^{2} \)
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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

−13.80832658042842592661127865303, −13.10227442338752365369864054415, −11.44557043877088879128765738433, −10.57440578657735152882540925382, −8.827104062926329491716274929933, −8.064968574265870627198111806531, −6.94645460846324809154954564180, −5.92607240875110110145163035708, −4.66902844881642411729754878148, −2.59825336491970313666444998998, 2.21996412766724760751917484491, 3.55477354531954723538833763218, 4.70163256403741205678875710037, 6.62837148358234187458345868320, 7.963877643924952661272178780234, 9.457492555984399402444689845909, 10.14941145070450050870183525646, 11.20009191630321018153376520163, 12.40630134502689394687409331838, 12.84160533280932639723836506271

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