Properties

Label 2-5e2-25.4-c5-0-0
Degree $2$
Conductor $25$
Sign $-0.996 + 0.0806i$
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  + (2.22 + 0.721i)2-s + (−9.09 + 12.5i)3-s + (−21.4 − 15.6i)4-s + (−55.6 − 4.76i)5-s + (−29.2 + 21.2i)6-s − 43.5i·7-s + (−80.3 − 110. i)8-s + (1.12 + 3.45i)9-s + (−120. − 50.7i)10-s + (−175. + 539. i)11-s + (390. − 126. i)12-s + (−283. + 92.1i)13-s + (31.4 − 96.7i)14-s + (566. − 653. i)15-s + (163. + 504. i)16-s + (−404. − 556. i)17-s + ⋯
L(s)  = 1  + (0.392 + 0.127i)2-s + (−0.583 + 0.802i)3-s + (−0.671 − 0.487i)4-s + (−0.996 − 0.0851i)5-s + (−0.331 + 0.240i)6-s − 0.335i·7-s + (−0.443 − 0.611i)8-s + (0.00461 + 0.0142i)9-s + (−0.380 − 0.160i)10-s + (−0.436 + 1.34i)11-s + (0.783 − 0.254i)12-s + (−0.465 + 0.151i)13-s + (0.0428 − 0.131i)14-s + (0.649 − 0.750i)15-s + (0.159 + 0.492i)16-s + (−0.339 − 0.467i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0806i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0806i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $-0.996 + 0.0806i$
Analytic conductor: \(4.00959\)
Root analytic conductor: \(2.00239\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{25} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 25,\ (\ :5/2),\ -0.996 + 0.0806i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.00884859 - 0.219036i\)
\(L(\frac12)\) \(\approx\) \(0.00884859 - 0.219036i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (55.6 + 4.76i)T \)
good2 \( 1 + (-2.22 - 0.721i)T + (25.8 + 18.8i)T^{2} \)
3 \( 1 + (9.09 - 12.5i)T + (-75.0 - 231. i)T^{2} \)
7 \( 1 + 43.5iT - 1.68e4T^{2} \)
11 \( 1 + (175. - 539. i)T + (-1.30e5 - 9.46e4i)T^{2} \)
13 \( 1 + (283. - 92.1i)T + (3.00e5 - 2.18e5i)T^{2} \)
17 \( 1 + (404. + 556. i)T + (-4.38e5 + 1.35e6i)T^{2} \)
19 \( 1 + (-912. + 663. i)T + (7.65e5 - 2.35e6i)T^{2} \)
23 \( 1 + (-19.1 - 6.20i)T + (5.20e6 + 3.78e6i)T^{2} \)
29 \( 1 + (5.68e3 + 4.13e3i)T + (6.33e6 + 1.95e7i)T^{2} \)
31 \( 1 + (997. - 724. i)T + (8.84e6 - 2.72e7i)T^{2} \)
37 \( 1 + (1.20e4 - 3.92e3i)T + (5.61e7 - 4.07e7i)T^{2} \)
41 \( 1 + (-5.77e3 - 1.77e4i)T + (-9.37e7 + 6.80e7i)T^{2} \)
43 \( 1 + 1.44e4iT - 1.47e8T^{2} \)
47 \( 1 + (-3.65e3 + 5.03e3i)T + (-7.08e7 - 2.18e8i)T^{2} \)
53 \( 1 + (1.23e4 - 1.69e4i)T + (-1.29e8 - 3.97e8i)T^{2} \)
59 \( 1 + (-1.01e4 - 3.13e4i)T + (-5.78e8 + 4.20e8i)T^{2} \)
61 \( 1 + (3.78e3 - 1.16e4i)T + (-6.83e8 - 4.96e8i)T^{2} \)
67 \( 1 + (3.55e4 + 4.88e4i)T + (-4.17e8 + 1.28e9i)T^{2} \)
71 \( 1 + (1.41e4 + 1.02e4i)T + (5.57e8 + 1.71e9i)T^{2} \)
73 \( 1 + (-1.97e4 - 6.43e3i)T + (1.67e9 + 1.21e9i)T^{2} \)
79 \( 1 + (5.17e4 + 3.75e4i)T + (9.50e8 + 2.92e9i)T^{2} \)
83 \( 1 + (-4.91e4 - 6.76e4i)T + (-1.21e9 + 3.74e9i)T^{2} \)
89 \( 1 + (3.87e4 - 1.19e5i)T + (-4.51e9 - 3.28e9i)T^{2} \)
97 \( 1 + (-3.87e4 + 5.33e4i)T + (-2.65e9 - 8.16e9i)T^{2} \)
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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

−16.98918951663329630289729830515, −15.65505878381416198570299840206, −15.03181149906796707985899827606, −13.43887049649749030698859887990, −12.02465973767375917293713691910, −10.57926777083068836539287879495, −9.439169622555459796182972556486, −7.36870564319467898493617075352, −5.10298835839231057101787578561, −4.22651034330342771256982619455, 0.14133218186573710357391492505, 3.53184511553169946838086457582, 5.55902674081827311861910914764, 7.47509479267269155680177678890, 8.765407860471369237760296474397, 11.16518164987775312294672542062, 12.20736869823765824705428916397, 13.01616586138306402819762653156, 14.45321086039992044771307084780, 15.97636761469435956084776672123

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