Properties

Label 2-5e2-5.3-c8-0-9
Degree $2$
Conductor $25$
Sign $-0.437 + 0.899i$
Analytic cond. $10.1844$
Root an. cond. $3.19131$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (21.8 − 21.8i)2-s + (42.5 + 42.5i)3-s − 697. i·4-s + 1.85e3·6-s + (432. − 432. i)7-s + (−9.64e3 − 9.64e3i)8-s − 2.94e3i·9-s + 5.24e3·11-s + (2.96e4 − 2.96e4i)12-s + (2.09e4 + 2.09e4i)13-s − 1.88e4i·14-s − 2.42e5·16-s + (−141. + 141. i)17-s + (−6.43e4 − 6.43e4i)18-s + 2.08e5i·19-s + ⋯
L(s)  = 1  + (1.36 − 1.36i)2-s + (0.524 + 0.524i)3-s − 2.72i·4-s + 1.43·6-s + (0.180 − 0.180i)7-s + (−2.35 − 2.35i)8-s − 0.449i·9-s + 0.358·11-s + (1.43 − 1.43i)12-s + (0.735 + 0.735i)13-s − 0.491i·14-s − 3.70·16-s + (−0.00168 + 0.00168i)17-s + (−0.613 − 0.613i)18-s + 1.59i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.437 + 0.899i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.437 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $-0.437 + 0.899i$
Analytic conductor: \(10.1844\)
Root analytic conductor: \(3.19131\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{25} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 25,\ (\ :4),\ -0.437 + 0.899i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(2.12941 - 3.40440i\)
\(L(\frac12)\) \(\approx\) \(2.12941 - 3.40440i\)
\(L(5)\) 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 \( 1 + (-21.8 + 21.8i)T - 256iT^{2} \)
3 \( 1 + (-42.5 - 42.5i)T + 6.56e3iT^{2} \)
7 \( 1 + (-432. + 432. i)T - 5.76e6iT^{2} \)
11 \( 1 - 5.24e3T + 2.14e8T^{2} \)
13 \( 1 + (-2.09e4 - 2.09e4i)T + 8.15e8iT^{2} \)
17 \( 1 + (141. - 141. i)T - 6.97e9iT^{2} \)
19 \( 1 - 2.08e5iT - 1.69e10T^{2} \)
23 \( 1 + (-6.23e3 - 6.23e3i)T + 7.83e10iT^{2} \)
29 \( 1 - 9.85e5iT - 5.00e11T^{2} \)
31 \( 1 - 4.14e5T + 8.52e11T^{2} \)
37 \( 1 + (-1.70e6 + 1.70e6i)T - 3.51e12iT^{2} \)
41 \( 1 - 1.82e6T + 7.98e12T^{2} \)
43 \( 1 + (2.29e6 + 2.29e6i)T + 1.16e13iT^{2} \)
47 \( 1 + (3.04e6 - 3.04e6i)T - 2.38e13iT^{2} \)
53 \( 1 + (5.37e6 + 5.37e6i)T + 6.22e13iT^{2} \)
59 \( 1 + 7.93e6iT - 1.46e14T^{2} \)
61 \( 1 + 1.12e7T + 1.91e14T^{2} \)
67 \( 1 + (1.52e7 - 1.52e7i)T - 4.06e14iT^{2} \)
71 \( 1 + 3.52e7T + 6.45e14T^{2} \)
73 \( 1 + (1.13e7 + 1.13e7i)T + 8.06e14iT^{2} \)
79 \( 1 - 3.50e6iT - 1.51e15T^{2} \)
83 \( 1 + (3.64e7 + 3.64e7i)T + 2.25e15iT^{2} \)
89 \( 1 - 7.49e7iT - 3.93e15T^{2} \)
97 \( 1 + (-9.73e7 + 9.73e7i)T - 7.83e15iT^{2} \)
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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

−14.63132446084602033234297178955, −14.14699000532994377678326061276, −12.71059403772845467587501955628, −11.59857525158582852845186212902, −10.36168693118400840669629958552, −9.157438743702339973951548180178, −6.13412665872032368276459751264, −4.33175618042393181406750696821, −3.33677909091646946773853213124, −1.46146746178224041004844241166, 2.87127622518831305225756984507, 4.68747022969024892571537676069, 6.22529847036377069543672514412, 7.57807348478089785401876417527, 8.589989528708373741266389208843, 11.55835181552450747641851526541, 13.10507264721413062215072659094, 13.59048455535989989512181612875, 14.85081042354239683060933029411, 15.73834428882758981975912237946

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