Properties

Label 2-5e2-25.11-c5-0-8
Degree $2$
Conductor $25$
Sign $0.195 + 0.980i$
Analytic cond. $4.00959$
Root an. cond. $2.00239$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (7.31 − 5.31i)2-s + (−0.760 + 2.33i)3-s + (15.4 − 47.3i)4-s + (−10.6 − 54.8i)5-s + (6.87 + 21.1i)6-s + 88.6·7-s + (−49.8 − 153. i)8-s + (191. + 139. i)9-s + (−370. − 344. i)10-s + (−521. + 379. i)11-s + (99.1 + 72.0i)12-s + (47.0 + 34.1i)13-s + (648. − 471. i)14-s + (136. + 16.6i)15-s + (109. + 79.4i)16-s + (331. + 1.02e3i)17-s + ⋯
L(s)  = 1  + (1.29 − 0.939i)2-s + (−0.0487 + 0.150i)3-s + (0.481 − 1.48i)4-s + (−0.191 − 0.981i)5-s + (0.0779 + 0.240i)6-s + 0.683·7-s + (−0.275 − 0.847i)8-s + (0.788 + 0.573i)9-s + (−1.17 − 1.08i)10-s + (−1.29 + 0.944i)11-s + (0.198 + 0.144i)12-s + (0.0771 + 0.0560i)13-s + (0.884 − 0.642i)14-s + (0.156 + 0.0191i)15-s + (0.106 + 0.0775i)16-s + (0.278 + 0.856i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.195 + 0.980i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.195 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $0.195 + 0.980i$
Analytic conductor: \(4.00959\)
Root analytic conductor: \(2.00239\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{25} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 25,\ (\ :5/2),\ 0.195 + 0.980i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.08950 - 1.71420i\)
\(L(\frac12)\) \(\approx\) \(2.08950 - 1.71420i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (10.6 + 54.8i)T \)
good2 \( 1 + (-7.31 + 5.31i)T + (9.88 - 30.4i)T^{2} \)
3 \( 1 + (0.760 - 2.33i)T + (-196. - 142. i)T^{2} \)
7 \( 1 - 88.6T + 1.68e4T^{2} \)
11 \( 1 + (521. - 379. i)T + (4.97e4 - 1.53e5i)T^{2} \)
13 \( 1 + (-47.0 - 34.1i)T + (1.14e5 + 3.53e5i)T^{2} \)
17 \( 1 + (-331. - 1.02e3i)T + (-1.14e6 + 8.34e5i)T^{2} \)
19 \( 1 + (362. + 1.11e3i)T + (-2.00e6 + 1.45e6i)T^{2} \)
23 \( 1 + (-541. + 393. i)T + (1.98e6 - 6.12e6i)T^{2} \)
29 \( 1 + (-656. + 2.01e3i)T + (-1.65e7 - 1.20e7i)T^{2} \)
31 \( 1 + (1.84e3 + 5.69e3i)T + (-2.31e7 + 1.68e7i)T^{2} \)
37 \( 1 + (9.03e3 + 6.56e3i)T + (2.14e7 + 6.59e7i)T^{2} \)
41 \( 1 + (-1.44e4 - 1.04e4i)T + (3.58e7 + 1.10e8i)T^{2} \)
43 \( 1 + 2.19e4T + 1.47e8T^{2} \)
47 \( 1 + (2.82e3 - 8.69e3i)T + (-1.85e8 - 1.34e8i)T^{2} \)
53 \( 1 + (-1.44e3 + 4.43e3i)T + (-3.38e8 - 2.45e8i)T^{2} \)
59 \( 1 + (1.22e4 + 8.88e3i)T + (2.20e8 + 6.79e8i)T^{2} \)
61 \( 1 + (-3.76e4 + 2.73e4i)T + (2.60e8 - 8.03e8i)T^{2} \)
67 \( 1 + (-1.82e4 - 5.61e4i)T + (-1.09e9 + 7.93e8i)T^{2} \)
71 \( 1 + (1.06e4 - 3.28e4i)T + (-1.45e9 - 1.06e9i)T^{2} \)
73 \( 1 + (2.50e4 - 1.81e4i)T + (6.40e8 - 1.97e9i)T^{2} \)
79 \( 1 + (-2.74e4 + 8.44e4i)T + (-2.48e9 - 1.80e9i)T^{2} \)
83 \( 1 + (-5.16e3 - 1.58e4i)T + (-3.18e9 + 2.31e9i)T^{2} \)
89 \( 1 + (-8.33e4 + 6.05e4i)T + (1.72e9 - 5.31e9i)T^{2} \)
97 \( 1 + (-2.60e4 + 8.02e4i)T + (-6.94e9 - 5.04e9i)T^{2} \)
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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

−15.87931402457231279523567637302, −14.82577050274405763796591054337, −13.17940287257147853789450084136, −12.75240024752148694896080985554, −11.32451220076186738062260784491, −10.11016554333512266158385397745, −7.946702849427919288617923976635, −5.20525225224142046655470259341, −4.35940380283678731536510856907, −1.91496062610742696968045619831, 3.38686946219453087001953293481, 5.25101959333006923351227308037, 6.75880890119395468698521716532, 7.893229364705455603923929298362, 10.50651071753815344793944403145, 12.03721725316891470794472229516, 13.40502874904748946400266408526, 14.36845175306810009221524510264, 15.37866074829472514755388297313, 16.23011625153665004992905121091

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