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Properties

Label 2-5e4-25.6-c1-0-7
Degree $2$
Conductor $625$
Sign $-0.146 + 0.989i$
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  + (−0.718 − 2.21i)2-s + (−1.86 + 1.35i)3-s + (−2.75 + 2.00i)4-s + (4.33 + 3.14i)6-s − 3.59·7-s + (2.65 + 1.92i)8-s + (0.710 − 2.18i)9-s + (−0.153 − 0.473i)11-s + (2.42 − 7.46i)12-s + (−0.818 + 2.51i)13-s + (2.58 + 7.95i)14-s + (0.250 − 0.771i)16-s + (4.13 + 3.00i)17-s − 5.35·18-s + (0.798 + 0.580i)19-s + ⋯
L(s)  = 1  + (−0.508 − 1.56i)2-s + (−1.07 + 0.781i)3-s + (−1.37 + 1.00i)4-s + (1.76 + 1.28i)6-s − 1.35·7-s + (0.938 + 0.681i)8-s + (0.236 − 0.729i)9-s + (−0.0463 − 0.142i)11-s + (0.700 − 2.15i)12-s + (−0.226 + 0.698i)13-s + (0.690 + 2.12i)14-s + (0.0627 − 0.192i)16-s + (1.00 + 0.728i)17-s − 1.26·18-s + (0.183 + 0.133i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.146 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.146 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(625\)    =    \(5^{4}\)
Sign: $-0.146 + 0.989i$
Analytic conductor: \(4.99065\)
Root analytic conductor: \(2.23397\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{625} (126, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 625,\ (\ :1/2),\ -0.146 + 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.289667 - 0.335583i\)
\(L(\frac12)\) \(\approx\) \(0.289667 - 0.335583i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.718 + 2.21i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (1.86 - 1.35i)T + (0.927 - 2.85i)T^{2} \)
7 \( 1 + 3.59T + 7T^{2} \)
11 \( 1 + (0.153 + 0.473i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (0.818 - 2.51i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-4.13 - 3.00i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.798 - 0.580i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (1.98 + 6.09i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-4.50 + 3.27i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (4.89 + 3.55i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-1.42 + 4.37i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.888 - 2.73i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 9.48T + 43T^{2} \)
47 \( 1 + (-4.34 + 3.15i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (0.248 - 0.180i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.391 - 1.20i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (1.92 + 5.92i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (4.27 + 3.10i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (-0.122 + 0.0893i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-4.59 - 14.1i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-13.4 + 9.75i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-11.8 - 8.57i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-3.51 - 10.8i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-0.687 + 0.499i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.53464412852672520614318927684, −9.714461199203826488688974834190, −9.333719147164002140563703420815, −8.081271280992999128976894754185, −6.52805760550189782968373890161, −5.74205135405605082174484614607, −4.34968282069563998708424365312, −3.64094241610560098932926303500, −2.40763538325450860008928193306, −0.52374286363108390194574204620, 0.77863013848943149582561579523, 3.21551117405045662126972490302, 5.10750564846443115560656364740, 5.76564803673611194712482770491, 6.41926208127287020411785347010, 7.26681650399969752644119914587, 7.67614627484377190320525785107, 9.054973195538249007446893069008, 9.709626653346970962683763426544, 10.61956172528841008436233105307

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