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Properties

Label 2-5e2-5.2-c6-0-5
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  + (2.71 + 2.71i)2-s + (16.9 − 16.9i)3-s − 49.2i·4-s + 92.2·6-s + (−383. − 383. i)7-s + (307. − 307. i)8-s + 152. i·9-s + 2.40e3·11-s + (−836. − 836. i)12-s + (332. − 332. i)13-s − 2.08e3i·14-s − 1.48e3·16-s + (373. + 373. i)17-s + (−412. + 412. i)18-s + 7.03e3i·19-s + ⋯
L(s)  = 1  + (0.339 + 0.339i)2-s + (0.629 − 0.629i)3-s − 0.769i·4-s + 0.427·6-s + (−1.11 − 1.11i)7-s + (0.600 − 0.600i)8-s + 0.208i·9-s + 1.80·11-s + (−0.484 − 0.484i)12-s + (0.151 − 0.151i)13-s − 0.759i·14-s − 0.361·16-s + (0.0760 + 0.0760i)17-s + (−0.0707 + 0.0707i)18-s + 1.02i·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

\(L(\frac{7}{2})\) \(\approx\) \(1.80689 - 1.13019i\)
\(L(\frac12)\) \(\approx\) \(1.80689 - 1.13019i\)
\(L(4)\) 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 \( 1 + (-2.71 - 2.71i)T + 64iT^{2} \)
3 \( 1 + (-16.9 + 16.9i)T - 729iT^{2} \)
7 \( 1 + (383. + 383. i)T + 1.17e5iT^{2} \)
11 \( 1 - 2.40e3T + 1.77e6T^{2} \)
13 \( 1 + (-332. + 332. i)T - 4.82e6iT^{2} \)
17 \( 1 + (-373. - 373. i)T + 2.41e7iT^{2} \)
19 \( 1 - 7.03e3iT - 4.70e7T^{2} \)
23 \( 1 + (2.61e3 - 2.61e3i)T - 1.48e8iT^{2} \)
29 \( 1 - 1.36e4iT - 5.94e8T^{2} \)
31 \( 1 - 1.07e4T + 8.87e8T^{2} \)
37 \( 1 + (2.99e4 + 2.99e4i)T + 2.56e9iT^{2} \)
41 \( 1 - 6.03e4T + 4.75e9T^{2} \)
43 \( 1 + (-6.38e4 + 6.38e4i)T - 6.32e9iT^{2} \)
47 \( 1 + (-7.43e4 - 7.43e4i)T + 1.07e10iT^{2} \)
53 \( 1 + (1.25e5 - 1.25e5i)T - 2.21e10iT^{2} \)
59 \( 1 + 1.52e5iT - 4.21e10T^{2} \)
61 \( 1 + 1.34e5T + 5.15e10T^{2} \)
67 \( 1 + (-6.90e4 - 6.90e4i)T + 9.04e10iT^{2} \)
71 \( 1 + 9.61e4T + 1.28e11T^{2} \)
73 \( 1 + (-2.31e5 + 2.31e5i)T - 1.51e11iT^{2} \)
79 \( 1 - 2.81e4iT - 2.43e11T^{2} \)
83 \( 1 + (5.49e5 - 5.49e5i)T - 3.26e11iT^{2} \)
89 \( 1 + 1.19e6iT - 4.96e11T^{2} \)
97 \( 1 + (-4.69e5 - 4.69e5i)T + 8.32e11iT^{2} \)
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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

−16.00608209142159549213048590359, −14.31969620660356902009280880334, −13.89383253724418506064865350265, −12.61804192393305451622410238432, −10.60023559490217574932361444881, −9.317263821266561506861559010442, −7.32345295788439426667361550663, −6.25571370892031099511169605180, −3.89013370680782036292210412741, −1.25165173809262749239888439028, 2.84724527591889659009133171278, 4.06497223377640138429993403034, 6.52225920956419586112800239273, 8.759879072773892671711783856103, 9.478753704732632725781467110255, 11.64176822172687406453174680904, 12.52802158882921132621272852878, 13.95930869155574276442624856711, 15.24211861614758397958175714458, 16.30023501451669856152724518831

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