Properties

Degree $2$
Conductor $25$
Sign $0.447 + 0.894i$
Motivic weight $33$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9.61e4i·2-s + 1.07e8i·3-s − 6.61e8·4-s − 1.03e13·6-s + 8.61e13i·7-s + 7.62e14i·8-s − 5.94e15·9-s + 9.57e16·11-s − 7.09e16i·12-s + 2.44e18i·13-s − 8.28e18·14-s − 7.90e19·16-s + 6.06e19i·17-s − 5.72e20i·18-s + 1.50e21·19-s + ⋯
L(s)  = 1  + 1.03i·2-s + 1.43i·3-s − 0.0770·4-s − 1.49·6-s + 0.979i·7-s + 0.957i·8-s − 1.07·9-s + 0.628·11-s − 0.110i·12-s + 1.01i·13-s − 1.01·14-s − 1.07·16-s + 0.302i·17-s − 1.11i·18-s + 1.19·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $0.447 + 0.894i$
Motivic weight: \(33\)
Character: $\chi_{25} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 25,\ (\ :33/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(17)\) \(\approx\) \(2.490702873\)
\(L(\frac12)\) \(\approx\) \(2.490702873\)
\(L(\frac{35}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
good2 \( 1 - 9.61e4iT - 8.58e9T^{2} \)
3 \( 1 - 1.07e8iT - 5.55e15T^{2} \)
7 \( 1 - 8.61e13iT - 7.73e27T^{2} \)
11 \( 1 - 9.57e16T + 2.32e34T^{2} \)
13 \( 1 - 2.44e18iT - 5.75e36T^{2} \)
17 \( 1 - 6.06e19iT - 4.02e40T^{2} \)
19 \( 1 - 1.50e21T + 1.58e42T^{2} \)
23 \( 1 - 1.88e22iT - 8.65e44T^{2} \)
29 \( 1 + 2.67e24T + 1.81e48T^{2} \)
31 \( 1 - 2.64e24T + 1.64e49T^{2} \)
37 \( 1 - 1.12e26iT - 5.63e51T^{2} \)
41 \( 1 - 7.31e26T + 1.66e53T^{2} \)
43 \( 1 - 3.88e26iT - 8.02e53T^{2} \)
47 \( 1 + 3.28e27iT - 1.51e55T^{2} \)
53 \( 1 - 3.35e28iT - 7.96e56T^{2} \)
59 \( 1 + 2.93e29T + 2.74e58T^{2} \)
61 \( 1 - 7.07e28T + 8.23e58T^{2} \)
67 \( 1 + 1.99e30iT - 1.82e60T^{2} \)
71 \( 1 + 1.15e30T + 1.23e61T^{2} \)
73 \( 1 + 2.83e30iT - 3.08e61T^{2} \)
79 \( 1 + 6.93e30T + 4.18e62T^{2} \)
83 \( 1 - 4.09e31iT - 2.13e63T^{2} \)
89 \( 1 - 1.49e32T + 2.13e64T^{2} \)
97 \( 1 + 1.00e33iT - 3.65e65T^{2} \)
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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

−11.89148372623931413096133782325, −11.10441119711021982114439574116, −9.510584963139215969758744921075, −8.978669843315431297581910020923, −7.57406989817797708719321231103, −6.19867848968370596877154420731, −5.36057878327032494611960430106, −4.34542169844798448064825188323, −3.10653806983231862804172490171, −1.72488636450738647619483916293, 0.52444599838672861662399858791, 0.925547184565010903530651526682, 1.89562006490296423106431805270, 2.94017429209892377489670225941, 4.02277680939894644556307118710, 5.88684882546226915763405002032, 7.12133012371193570551222011996, 7.65524145068592017204525345952, 9.417284450743140616372791548914, 10.66897344718016034019936919694

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