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Properties

Label 2-5e2-5.3-c8-0-3
Degree $2$
Conductor $25$
Sign $0.850 - 0.525i$
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  + (4.39 − 4.39i)2-s + (−75.2 − 75.2i)3-s + 217. i·4-s − 662.·6-s + (−730. + 730. i)7-s + (2.08e3 + 2.08e3i)8-s + 4.77e3i·9-s + 1.95e4·11-s + (1.63e4 − 1.63e4i)12-s + (2.49e4 + 2.49e4i)13-s + 6.42e3i·14-s − 3.73e4·16-s + (1.12e4 − 1.12e4i)17-s + (2.10e4 + 2.10e4i)18-s + 1.71e5i·19-s + ⋯
L(s)  = 1  + (0.274 − 0.274i)2-s + (−0.929 − 0.929i)3-s + 0.849i·4-s − 0.510·6-s + (−0.304 + 0.304i)7-s + (0.508 + 0.508i)8-s + 0.728i·9-s + 1.33·11-s + (0.789 − 0.789i)12-s + (0.872 + 0.872i)13-s + 0.167i·14-s − 0.569·16-s + (0.135 − 0.135i)17-s + (0.200 + 0.200i)18-s + 1.31i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $0.850 - 0.525i$
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: \((2,\ 25,\ (\ :4),\ 0.850 - 0.525i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.33379 + 0.378904i\)
\(L(\frac12)\) \(\approx\) \(1.33379 + 0.378904i\)
\(L(5)\) 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 \( 1 + (-4.39 + 4.39i)T - 256iT^{2} \)
3 \( 1 + (75.2 + 75.2i)T + 6.56e3iT^{2} \)
7 \( 1 + (730. - 730. i)T - 5.76e6iT^{2} \)
11 \( 1 - 1.95e4T + 2.14e8T^{2} \)
13 \( 1 + (-2.49e4 - 2.49e4i)T + 8.15e8iT^{2} \)
17 \( 1 + (-1.12e4 + 1.12e4i)T - 6.97e9iT^{2} \)
19 \( 1 - 1.71e5iT - 1.69e10T^{2} \)
23 \( 1 + (-1.32e5 - 1.32e5i)T + 7.83e10iT^{2} \)
29 \( 1 - 1.27e5iT - 5.00e11T^{2} \)
31 \( 1 + 9.60e5T + 8.52e11T^{2} \)
37 \( 1 + (-2.43e5 + 2.43e5i)T - 3.51e12iT^{2} \)
41 \( 1 - 2.50e6T + 7.98e12T^{2} \)
43 \( 1 + (-6.76e3 - 6.76e3i)T + 1.16e13iT^{2} \)
47 \( 1 + (-1.79e6 + 1.79e6i)T - 2.38e13iT^{2} \)
53 \( 1 + (-2.97e6 - 2.97e6i)T + 6.22e13iT^{2} \)
59 \( 1 + 3.13e5iT - 1.46e14T^{2} \)
61 \( 1 - 1.76e7T + 1.91e14T^{2} \)
67 \( 1 + (4.41e6 - 4.41e6i)T - 4.06e14iT^{2} \)
71 \( 1 + 8.89e6T + 6.45e14T^{2} \)
73 \( 1 + (1.95e7 + 1.95e7i)T + 8.06e14iT^{2} \)
79 \( 1 - 1.11e7iT - 1.51e15T^{2} \)
83 \( 1 + (1.58e7 + 1.58e7i)T + 2.25e15iT^{2} \)
89 \( 1 + 4.85e7iT - 3.93e15T^{2} \)
97 \( 1 + (-1.07e8 + 1.07e8i)T - 7.83e15iT^{2} \)
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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

−16.32629322895145909505603436191, −14.22597322679657479461642223998, −12.92574316951554297403222941477, −12.02504232126976264003380163469, −11.28266774480938344858038193724, −8.985335390637238358964973044113, −7.24469024020373587475844577141, −6.02639631186924689094996340606, −3.81200992741517452995036326414, −1.51791422677900550244270476584, 0.75043457737565658007698792307, 4.09557890360841922166127195116, 5.47467305769941621393803982258, 6.65441055594003653837770060219, 9.256476683093154628116969632934, 10.49139343830622792779583102586, 11.34522605248965603547364656575, 13.23037904186019588483393518567, 14.66321581147272136325234620407, 15.69188106553845680402700726898

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