Properties

Label 2-5e2-25.11-c3-0-5
Degree $2$
Conductor $25$
Sign $0.778 + 0.627i$
Analytic cond. $1.47504$
Root an. cond. $1.21451$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.29 − 3.12i)2-s + (−1.93 + 5.96i)3-s + (6.24 − 19.2i)4-s + (−9.58 + 5.75i)5-s + (10.2 + 31.6i)6-s − 9.63·7-s + (−20.0 − 61.6i)8-s + (−9.96 − 7.24i)9-s + (−23.2 + 54.6i)10-s + (19.6 − 14.3i)11-s + (102. + 74.5i)12-s + (35.7 + 25.9i)13-s + (−41.4 + 30.0i)14-s + (−15.7 − 68.3i)15-s + (−147. − 107. i)16-s + (0.568 + 1.75i)17-s + ⋯
L(s)  = 1  + (1.51 − 1.10i)2-s + (−0.372 + 1.14i)3-s + (0.780 − 2.40i)4-s + (−0.857 + 0.514i)5-s + (0.700 + 2.15i)6-s − 0.520·7-s + (−0.885 − 2.72i)8-s + (−0.369 − 0.268i)9-s + (−0.734 + 1.72i)10-s + (0.539 − 0.392i)11-s + (2.46 + 1.79i)12-s + (0.762 + 0.554i)13-s + (−0.790 + 0.574i)14-s + (−0.271 − 1.17i)15-s + (−2.31 − 1.67i)16-s + (0.00811 + 0.0249i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.778 + 0.627i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.778 + 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $0.778 + 0.627i$
Analytic conductor: \(1.47504\)
Root analytic conductor: \(1.21451\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{25} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 25,\ (\ :3/2),\ 0.778 + 0.627i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.78313 - 0.628645i\)
\(L(\frac12)\) \(\approx\) \(1.78313 - 0.628645i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (9.58 - 5.75i)T \)
good2 \( 1 + (-4.29 + 3.12i)T + (2.47 - 7.60i)T^{2} \)
3 \( 1 + (1.93 - 5.96i)T + (-21.8 - 15.8i)T^{2} \)
7 \( 1 + 9.63T + 343T^{2} \)
11 \( 1 + (-19.6 + 14.3i)T + (411. - 1.26e3i)T^{2} \)
13 \( 1 + (-35.7 - 25.9i)T + (678. + 2.08e3i)T^{2} \)
17 \( 1 + (-0.568 - 1.75i)T + (-3.97e3 + 2.88e3i)T^{2} \)
19 \( 1 + (22.1 + 68.2i)T + (-5.54e3 + 4.03e3i)T^{2} \)
23 \( 1 + (56.7 - 41.2i)T + (3.75e3 - 1.15e4i)T^{2} \)
29 \( 1 + (31.0 - 95.6i)T + (-1.97e4 - 1.43e4i)T^{2} \)
31 \( 1 + (51.1 + 157. i)T + (-2.41e4 + 1.75e4i)T^{2} \)
37 \( 1 + (-316. - 229. i)T + (1.56e4 + 4.81e4i)T^{2} \)
41 \( 1 + (2.89 + 2.10i)T + (2.12e4 + 6.55e4i)T^{2} \)
43 \( 1 - 9.64T + 7.95e4T^{2} \)
47 \( 1 + (167. - 515. i)T + (-8.39e4 - 6.10e4i)T^{2} \)
53 \( 1 + (-50.2 + 154. i)T + (-1.20e5 - 8.75e4i)T^{2} \)
59 \( 1 + (134. + 97.6i)T + (6.34e4 + 1.95e5i)T^{2} \)
61 \( 1 + (-692. + 502. i)T + (7.01e4 - 2.15e5i)T^{2} \)
67 \( 1 + (218. + 671. i)T + (-2.43e5 + 1.76e5i)T^{2} \)
71 \( 1 + (209. - 644. i)T + (-2.89e5 - 2.10e5i)T^{2} \)
73 \( 1 + (-39.1 + 28.4i)T + (1.20e5 - 3.69e5i)T^{2} \)
79 \( 1 + (-145. + 448. i)T + (-3.98e5 - 2.89e5i)T^{2} \)
83 \( 1 + (185. + 572. i)T + (-4.62e5 + 3.36e5i)T^{2} \)
89 \( 1 + (558. - 406. i)T + (2.17e5 - 6.70e5i)T^{2} \)
97 \( 1 + (190. - 585. i)T + (-7.38e5 - 5.36e5i)T^{2} \)
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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

−16.28505007400702455548874497769, −15.45807441326804196444107902690, −14.44153671746692812386187070542, −13.06729314952174835661769250171, −11.50456241483284736638340075900, −11.01951176126481907243175054766, −9.660992224339289587414503109111, −6.28744435979204827396181593619, −4.47707200632211495859973607277, −3.44714522958065633275445861733, 3.94350988106668507160396590500, 5.88804508237748569112383263748, 7.05408625876975487585229422866, 8.194390632249457776132376931476, 11.72775495926396437540802892905, 12.57652551734478045448267600045, 13.24907530941615669634549381299, 14.70937758976228442054237029827, 15.90026168181467252044798300008, 16.77507525360369345537176054294

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