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Properties

Label 2-5e4-25.4-c1-0-12
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  + (−0.587 − 0.190i)2-s + (0.587 − 0.809i)3-s + (−1.30 − 0.951i)4-s + (−0.5 + 0.363i)6-s + 1.61i·7-s + (1.31 + 1.80i)8-s + (0.618 + 1.90i)9-s + (−0.236 + 0.726i)11-s + (−1.53 + 0.5i)12-s + (4.61 − 1.5i)13-s + (0.309 − 0.951i)14-s + (0.572 + 1.76i)16-s + (−0.449 − 0.618i)17-s − 1.23i·18-s + (−4.73 + 3.44i)19-s + ⋯
L(s)  = 1  + (−0.415 − 0.135i)2-s + (0.339 − 0.467i)3-s + (−0.654 − 0.475i)4-s + (−0.204 + 0.148i)6-s + 0.611i·7-s + (0.464 + 0.639i)8-s + (0.206 + 0.634i)9-s + (−0.0711 + 0.219i)11-s + (−0.444 + 0.144i)12-s + (1.28 − 0.416i)13-s + (0.0825 − 0.254i)14-s + (0.143 + 0.440i)16-s + (−0.108 − 0.149i)17-s − 0.291i·18-s + (−1.08 + 0.789i)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: $\chi_{625} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 625,\ (\ :1/2),\ 0.998 + 0.0627i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18830 - 0.0373441i\)
\(L(\frac12)\) \(\approx\) \(1.18830 - 0.0373441i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.587 + 0.190i)T + (1.61 + 1.17i)T^{2} \)
3 \( 1 + (-0.587 + 0.809i)T + (-0.927 - 2.85i)T^{2} \)
7 \( 1 - 1.61iT - 7T^{2} \)
11 \( 1 + (0.236 - 0.726i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-4.61 + 1.5i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.449 + 0.618i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (4.73 - 3.44i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-7.83 - 2.54i)T + (18.6 + 13.5i)T^{2} \)
29 \( 1 + (1.11 + 0.812i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-2.42 + 1.76i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-4.02 + 1.30i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.61 + 4.97i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 1.85iT - 43T^{2} \)
47 \( 1 + (-0.951 + 1.30i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-3.21 + 4.42i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.28 - 3.94i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (1.45 - 4.47i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-5.42 - 7.47i)T + (-20.7 + 63.7i)T^{2} \)
71 \( 1 + (-3.54 - 2.57i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (8.55 + 2.78i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-2.5 - 1.81i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-1.03 - 1.42i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + (2.76 - 8.50i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (1.67 - 2.30i)T + (-29.9 - 92.2i)T^{2} \)
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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

−10.61204653924956521002981288273, −9.655045656474259819963377767947, −8.654600664942010568288407471242, −8.333224966426115675529790602959, −7.23052195421831150335478908692, −5.99235637449388341146352829101, −5.17997921976860967020834854948, −4.00569653509453866624447546655, −2.44433168153135731208698323175, −1.26343213274245579410755474701, 0.919674443844184187233151908836, 3.11282067300901489233530388102, 4.03189173079629639045283845111, 4.71795664366210514403095336864, 6.37988421023158289601432562473, 7.09315724976389717767158205227, 8.350283386154226048067499400223, 8.834431420142707770352903014704, 9.517516200623471792672188156115, 10.53548074945752836602774513151

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