Properties

Label 2-5e2-5.3-c8-0-4
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  + (−15.2 + 15.2i)2-s + (20.3 + 20.3i)3-s − 209. i·4-s − 620.·6-s + (2.41e3 − 2.41e3i)7-s + (−705. − 705. i)8-s − 5.73e3i·9-s − 981.·11-s + (4.26e3 − 4.26e3i)12-s + (2.65e4 + 2.65e4i)13-s + 7.37e4i·14-s + 7.52e4·16-s + (1.85e4 − 1.85e4i)17-s + (8.75e4 + 8.75e4i)18-s + 5.03e4i·19-s + ⋯
L(s)  = 1  + (−0.953 + 0.953i)2-s + (0.251 + 0.251i)3-s − 0.819i·4-s − 0.478·6-s + (1.00 − 1.00i)7-s + (−0.172 − 0.172i)8-s − 0.873i·9-s − 0.0670·11-s + (0.205 − 0.205i)12-s + (0.930 + 0.930i)13-s + 1.91i·14-s + 1.14·16-s + (0.221 − 0.221i)17-s + (0.833 + 0.833i)18-s + 0.386i·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

\(L(\frac{9}{2})\) \(\approx\) \(1.18135 + 0.335597i\)
\(L(\frac12)\) \(\approx\) \(1.18135 + 0.335597i\)
\(L(5)\) 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 \( 1 + (15.2 - 15.2i)T - 256iT^{2} \)
3 \( 1 + (-20.3 - 20.3i)T + 6.56e3iT^{2} \)
7 \( 1 + (-2.41e3 + 2.41e3i)T - 5.76e6iT^{2} \)
11 \( 1 + 981.T + 2.14e8T^{2} \)
13 \( 1 + (-2.65e4 - 2.65e4i)T + 8.15e8iT^{2} \)
17 \( 1 + (-1.85e4 + 1.85e4i)T - 6.97e9iT^{2} \)
19 \( 1 - 5.03e4iT - 1.69e10T^{2} \)
23 \( 1 + (1.36e4 + 1.36e4i)T + 7.83e10iT^{2} \)
29 \( 1 + 1.05e6iT - 5.00e11T^{2} \)
31 \( 1 - 1.09e6T + 8.52e11T^{2} \)
37 \( 1 + (7.78e4 - 7.78e4i)T - 3.51e12iT^{2} \)
41 \( 1 - 5.54e5T + 7.98e12T^{2} \)
43 \( 1 + (-1.07e6 - 1.07e6i)T + 1.16e13iT^{2} \)
47 \( 1 + (-4.06e6 + 4.06e6i)T - 2.38e13iT^{2} \)
53 \( 1 + (1.88e6 + 1.88e6i)T + 6.22e13iT^{2} \)
59 \( 1 - 1.27e7iT - 1.46e14T^{2} \)
61 \( 1 - 1.40e7T + 1.91e14T^{2} \)
67 \( 1 + (-9.54e6 + 9.54e6i)T - 4.06e14iT^{2} \)
71 \( 1 + 2.82e7T + 6.45e14T^{2} \)
73 \( 1 + (1.11e7 + 1.11e7i)T + 8.06e14iT^{2} \)
79 \( 1 + 6.87e7iT - 1.51e15T^{2} \)
83 \( 1 + (-3.29e6 - 3.29e6i)T + 2.25e15iT^{2} \)
89 \( 1 - 7.97e7iT - 3.93e15T^{2} \)
97 \( 1 + (-1.96e7 + 1.96e7i)T - 7.83e15iT^{2} \)
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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

−16.02935643099005033785014650004, −14.88561632584650919936284754541, −13.78861901846853408692952815763, −11.73720522251153434076236379783, −10.13802455909784238398354644135, −8.844360779396286192898348308948, −7.71867691809076516456018797859, −6.36168151484725979527329270970, −4.03517237650346445156555441926, −0.932586909704866578302518666279, 1.34323841919092348692601390911, 2.67631470631891855067228268181, 5.42002089074731565115885395530, 8.029021769152928915372774382162, 8.789613669481220813234049439263, 10.49226224747224377778594777371, 11.39755441320694081854021850244, 12.72127825501834317571786738546, 14.33232271311888893991838088654, 15.68131443797347257887808065155

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