Properties

Degree $2$
Conductor $25$
Sign $0.525 - 0.850i$
Motivic weight $8$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−11.8 − 11.8i)2-s + (−11.8 + 11.8i)3-s + 26i·4-s + 282·6-s + (−2.74e3 − 2.74e3i)7-s + (−2.73e3 + 2.73e3i)8-s + 6.27e3i·9-s + 1.21e4·11-s + (−308. − 308. i)12-s + (2.42e3 − 2.42e3i)13-s + 6.51e4i·14-s + 7.15e4·16-s + (7.69e4 + 7.69e4i)17-s + (7.45e4 − 7.45e4i)18-s + 1.68e5i·19-s + ⋯
L(s)  = 1  + (−0.742 − 0.742i)2-s + (−0.146 + 0.146i)3-s + 0.101i·4-s + 0.217·6-s + (−1.14 − 1.14i)7-s + (−0.666 + 0.666i)8-s + 0.957i·9-s + 0.828·11-s + (−0.0148 − 0.0148i)12-s + (0.0848 − 0.0848i)13-s + 1.69i·14-s + 1.09·16-s + (0.921 + 0.921i)17-s + (0.710 − 0.710i)18-s + 1.29i·19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $0.525 - 0.850i$
Motivic weight: \(8\)
Character: $\chi_{25} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 25,\ (\ :4),\ 0.525 - 0.850i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.410624 + 0.228937i\)
\(L(\frac12)\) \(\approx\) \(0.410624 + 0.228937i\)
\(L(5)\) 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 \( 1 + (11.8 + 11.8i)T + 256iT^{2} \)
3 \( 1 + (11.8 - 11.8i)T - 6.56e3iT^{2} \)
7 \( 1 + (2.74e3 + 2.74e3i)T + 5.76e6iT^{2} \)
11 \( 1 - 1.21e4T + 2.14e8T^{2} \)
13 \( 1 + (-2.42e3 + 2.42e3i)T - 8.15e8iT^{2} \)
17 \( 1 + (-7.69e4 - 7.69e4i)T + 6.97e9iT^{2} \)
19 \( 1 - 1.68e5iT - 1.69e10T^{2} \)
23 \( 1 + (2.21e5 - 2.21e5i)T - 7.83e10iT^{2} \)
29 \( 1 + 6.66e5iT - 5.00e11T^{2} \)
31 \( 1 + 1.04e6T + 8.52e11T^{2} \)
37 \( 1 + (-2.07e6 - 2.07e6i)T + 3.51e12iT^{2} \)
41 \( 1 + 1.32e6T + 7.98e12T^{2} \)
43 \( 1 + (2.78e6 - 2.78e6i)T - 1.16e13iT^{2} \)
47 \( 1 + (3.63e6 + 3.63e6i)T + 2.38e13iT^{2} \)
53 \( 1 + (-3.14e6 + 3.14e6i)T - 6.22e13iT^{2} \)
59 \( 1 - 6.49e6iT - 1.46e14T^{2} \)
61 \( 1 + 1.43e7T + 1.91e14T^{2} \)
67 \( 1 + (-1.14e7 - 1.14e7i)T + 4.06e14iT^{2} \)
71 \( 1 + 2.30e7T + 6.45e14T^{2} \)
73 \( 1 + (1.74e7 - 1.74e7i)T - 8.06e14iT^{2} \)
79 \( 1 + 2.76e6iT - 1.51e15T^{2} \)
83 \( 1 + (-1.15e7 + 1.15e7i)T - 2.25e15iT^{2} \)
89 \( 1 - 2.61e7iT - 3.93e15T^{2} \)
97 \( 1 + (-8.09e7 - 8.09e7i)T + 7.83e15iT^{2} \)
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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

−16.43490742455374953569668882001, −14.59745057270789020047838245232, −13.28359795662070995274765461016, −11.73445606518952973722556773680, −10.33562561084523302323539833859, −9.804950947862602499730862019986, −7.950051031464391513647461378696, −6.02492577982123154300267029597, −3.65382816190440702852192984635, −1.44747688200536942416589873753, 0.30000086782784413153038542258, 3.23481496563591788452217803052, 6.03751932657559777528337007657, 7.02420414343425529715621808214, 8.940562126313141866949483484668, 9.503854729516589760068196366077, 11.90064796819903142855703005266, 12.71552419960476611938409235411, 14.72112562615102347436661237808, 15.84629049040102281132982324188

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