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Properties

Label 2-5e2-5.2-c6-0-1
Degree $2$
Conductor $25$
Sign $0.437 + 0.899i$
Analytic cond. $5.75135$
Root an. cond. $2.39819$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−8.83 − 8.83i)2-s + (−29.2 + 29.2i)3-s + 92.2i·4-s + 516.·6-s + (−106. − 106. i)7-s + (249. − 249. i)8-s − 980. i·9-s − 428.·11-s + (−2.69e3 − 2.69e3i)12-s + (2.82e3 − 2.82e3i)13-s + 1.87e3i·14-s + 1.48e3·16-s + (5.27e3 + 5.27e3i)17-s + (−8.66e3 + 8.66e3i)18-s − 3.72e3i·19-s + ⋯
L(s)  = 1  + (−1.10 − 1.10i)2-s + (−1.08 + 1.08i)3-s + 1.44i·4-s + 2.39·6-s + (−0.309 − 0.309i)7-s + (0.487 − 0.487i)8-s − 1.34i·9-s − 0.321·11-s + (−1.56 − 1.56i)12-s + (1.28 − 1.28i)13-s + 0.684i·14-s + 0.363·16-s + (1.07 + 1.07i)17-s + (−1.48 + 1.48i)18-s − 0.542i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.437 + 0.899i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.437 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $0.437 + 0.899i$
Analytic conductor: \(5.75135\)
Root analytic conductor: \(2.39819\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{25} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 25,\ (\ :3),\ 0.437 + 0.899i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.438372 - 0.274197i\)
\(L(\frac12)\) \(\approx\) \(0.438372 - 0.274197i\)
\(L(4)\) 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 \( 1 + (8.83 + 8.83i)T + 64iT^{2} \)
3 \( 1 + (29.2 - 29.2i)T - 729iT^{2} \)
7 \( 1 + (106. + 106. i)T + 1.17e5iT^{2} \)
11 \( 1 + 428.T + 1.77e6T^{2} \)
13 \( 1 + (-2.82e3 + 2.82e3i)T - 4.82e6iT^{2} \)
17 \( 1 + (-5.27e3 - 5.27e3i)T + 2.41e7iT^{2} \)
19 \( 1 + 3.72e3iT - 4.70e7T^{2} \)
23 \( 1 + (1.03e3 - 1.03e3i)T - 1.48e8iT^{2} \)
29 \( 1 + 1.08e3iT - 5.94e8T^{2} \)
31 \( 1 - 1.64e4T + 8.87e8T^{2} \)
37 \( 1 + (-2.58e4 - 2.58e4i)T + 2.56e9iT^{2} \)
41 \( 1 - 3.76e4T + 4.75e9T^{2} \)
43 \( 1 + (-4.97e4 + 4.97e4i)T - 6.32e9iT^{2} \)
47 \( 1 + (1.12e5 + 1.12e5i)T + 1.07e10iT^{2} \)
53 \( 1 + (-6.53e4 + 6.53e4i)T - 2.21e10iT^{2} \)
59 \( 1 - 1.45e5iT - 4.21e10T^{2} \)
61 \( 1 - 2.33e5T + 5.15e10T^{2} \)
67 \( 1 + (-7.96e4 - 7.96e4i)T + 9.04e10iT^{2} \)
71 \( 1 - 3.00e5T + 1.28e11T^{2} \)
73 \( 1 + (-1.45e5 + 1.45e5i)T - 1.51e11iT^{2} \)
79 \( 1 + 4.62e5iT - 2.43e11T^{2} \)
83 \( 1 + (5.43e5 - 5.43e5i)T - 3.26e11iT^{2} \)
89 \( 1 + 1.12e6iT - 4.96e11T^{2} \)
97 \( 1 + (8.40e5 + 8.40e5i)T + 8.32e11iT^{2} \)
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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

−16.46987054880005576049864376596, −15.29858986384498171408563687582, −12.91197268844151884413955679931, −11.51984373210028998610638668734, −10.53610009976504932242590852171, −9.985368860370607946313984274530, −8.327308380934951115860836847806, −5.71964029570979102332177548431, −3.54713813405353051018635990983, −0.68073656896958963096226372882, 1.02974670839191747743272192340, 5.78689948175829278255912095543, 6.69938808218772317291112227646, 7.922113575988214787767690590878, 9.483230734881251130936823561908, 11.25076304686193386034529024405, 12.53254604232911782900160656491, 14.11368327157775612809542198282, 16.00440503213869322198378429064, 16.53876196310176486685594005020

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