Properties

Label 2-5e2-25.13-c2-0-0
Degree $2$
Conductor $25$
Sign $-0.268 - 0.963i$
Analytic cond. $0.681200$
Root an. cond. $0.825348$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.86 + 0.295i)2-s + (−2.19 + 4.30i)3-s + (−0.405 + 0.131i)4-s + (4.99 − 0.209i)5-s + (2.82 − 8.69i)6-s + (−3.57 + 3.57i)7-s + (7.45 − 3.79i)8-s + (−8.44 − 11.6i)9-s + (−9.26 + 1.86i)10-s + (11.7 + 8.53i)11-s + (0.322 − 2.03i)12-s + (1.48 + 0.234i)13-s + (5.61 − 7.72i)14-s + (−10.0 + 21.9i)15-s + (−11.4 + 8.29i)16-s + (0.980 + 1.92i)17-s + ⋯
L(s)  = 1  + (−0.933 + 0.147i)2-s + (−0.731 + 1.43i)3-s + (−0.101 + 0.0329i)4-s + (0.999 − 0.0419i)5-s + (0.470 − 1.44i)6-s + (−0.510 + 0.510i)7-s + (0.931 − 0.474i)8-s + (−0.938 − 1.29i)9-s + (−0.926 + 0.186i)10-s + (1.06 + 0.776i)11-s + (0.0268 − 0.169i)12-s + (0.114 + 0.0180i)13-s + (0.400 − 0.551i)14-s + (−0.670 + 1.46i)15-s + (−0.713 + 0.518i)16-s + (0.0576 + 0.113i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.268 - 0.963i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.268 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $-0.268 - 0.963i$
Analytic conductor: \(0.681200\)
Root analytic conductor: \(0.825348\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{25} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 25,\ (\ :1),\ -0.268 - 0.963i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.309141 + 0.407301i\)
\(L(\frac12)\) \(\approx\) \(0.309141 + 0.407301i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-4.99 + 0.209i)T \)
good2 \( 1 + (1.86 - 0.295i)T + (3.80 - 1.23i)T^{2} \)
3 \( 1 + (2.19 - 4.30i)T + (-5.29 - 7.28i)T^{2} \)
7 \( 1 + (3.57 - 3.57i)T - 49iT^{2} \)
11 \( 1 + (-11.7 - 8.53i)T + (37.3 + 115. i)T^{2} \)
13 \( 1 + (-1.48 - 0.234i)T + (160. + 52.2i)T^{2} \)
17 \( 1 + (-0.980 - 1.92i)T + (-169. + 233. i)T^{2} \)
19 \( 1 + (0.665 + 0.216i)T + (292. + 212. i)T^{2} \)
23 \( 1 + (5.44 + 34.3i)T + (-503. + 163. i)T^{2} \)
29 \( 1 + (-23.5 + 7.64i)T + (680. - 494. i)T^{2} \)
31 \( 1 + (0.269 - 0.830i)T + (-777. - 564. i)T^{2} \)
37 \( 1 + (-3.63 + 22.9i)T + (-1.30e3 - 423. i)T^{2} \)
41 \( 1 + (17.9 - 13.0i)T + (519. - 1.59e3i)T^{2} \)
43 \( 1 + (-7.47 - 7.47i)T + 1.84e3iT^{2} \)
47 \( 1 + (-69.0 - 35.1i)T + (1.29e3 + 1.78e3i)T^{2} \)
53 \( 1 + (-11.3 + 22.3i)T + (-1.65e3 - 2.27e3i)T^{2} \)
59 \( 1 + (38.5 + 53.0i)T + (-1.07e3 + 3.31e3i)T^{2} \)
61 \( 1 + (71.2 + 51.7i)T + (1.14e3 + 3.53e3i)T^{2} \)
67 \( 1 + (40.4 + 79.4i)T + (-2.63e3 + 3.63e3i)T^{2} \)
71 \( 1 + (8.73 + 26.8i)T + (-4.07e3 + 2.96e3i)T^{2} \)
73 \( 1 + (-13.1 - 82.8i)T + (-5.06e3 + 1.64e3i)T^{2} \)
79 \( 1 + (102. - 33.3i)T + (5.04e3 - 3.66e3i)T^{2} \)
83 \( 1 + (22.5 - 11.4i)T + (4.04e3 - 5.57e3i)T^{2} \)
89 \( 1 + (-21.8 + 30.0i)T + (-2.44e3 - 7.53e3i)T^{2} \)
97 \( 1 + (-53.1 - 27.0i)T + (5.53e3 + 7.61e3i)T^{2} \)
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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

−17.39045793582760790121491633947, −16.82202957936817623566054251134, −15.76129956690790477499231483083, −14.30904902665665503317300903512, −12.45854313425682408749401634404, −10.61733643626499469224211977502, −9.700799445401889793577417582801, −8.985233585838164923059370551932, −6.30871362946539354612327930223, −4.51946622826956937814092887161, 1.24150016808955104390600430118, 5.89142886253702718005711281505, 7.19879687598290177044992151781, 8.932033104745545337016408762169, 10.36025834638082769190588553111, 11.76982003098744870589957220963, 13.39061968369439524946298368842, 13.89301999071141002478697131779, 16.70272844165463050815972835167, 17.29768694355859080949111280915

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