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  + 1.78e5i·2-s − 8.55e7i·3-s − 2.34e10·4-s + 1.52e13·6-s − 8.09e13i·7-s − 2.64e15i·8-s − 1.75e15·9-s + 1.95e17·11-s + 2.00e18i·12-s + 2.59e18i·13-s + 1.44e19·14-s + 2.72e20·16-s − 1.52e20i·17-s − 3.13e20i·18-s − 1.39e21·19-s + ⋯
L(s)  = 1  + 1.92i·2-s − 1.14i·3-s − 2.72·4-s + 2.21·6-s − 0.920i·7-s − 3.32i·8-s − 0.315·9-s + 1.28·11-s + 3.12i·12-s + 1.08i·13-s + 1.77·14-s + 3.69·16-s − 0.758i·17-s − 0.608i·18-s − 1.10·19-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

\(L(17)\) \(\approx\) \(1.820403983\)
\(L(\frac12)\) \(\approx\) \(1.820403983\)
\(L(\frac{35}{2})\) 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 \( 1 - 1.78e5iT - 8.58e9T^{2} \)
3 \( 1 + 8.55e7iT - 5.55e15T^{2} \)
7 \( 1 + 8.09e13iT - 7.73e27T^{2} \)
11 \( 1 - 1.95e17T + 2.32e34T^{2} \)
13 \( 1 - 2.59e18iT - 5.75e36T^{2} \)
17 \( 1 + 1.52e20iT - 4.02e40T^{2} \)
19 \( 1 + 1.39e21T + 1.58e42T^{2} \)
23 \( 1 - 3.40e22iT - 8.65e44T^{2} \)
29 \( 1 - 1.49e24T + 1.81e48T^{2} \)
31 \( 1 + 1.38e24T + 1.64e49T^{2} \)
37 \( 1 - 1.15e26iT - 5.63e51T^{2} \)
41 \( 1 + 6.35e25T + 1.66e53T^{2} \)
43 \( 1 - 2.64e26iT - 8.02e53T^{2} \)
47 \( 1 - 1.03e27iT - 1.51e55T^{2} \)
53 \( 1 + 5.20e28iT - 7.96e56T^{2} \)
59 \( 1 - 6.17e28T + 2.74e58T^{2} \)
61 \( 1 + 1.21e29T + 8.23e58T^{2} \)
67 \( 1 + 2.87e29iT - 1.82e60T^{2} \)
71 \( 1 + 2.84e30T + 1.23e61T^{2} \)
73 \( 1 - 2.67e30iT - 3.08e61T^{2} \)
79 \( 1 - 7.84e30T + 4.18e62T^{2} \)
83 \( 1 + 1.28e31iT - 2.13e63T^{2} \)
89 \( 1 - 1.56e32T + 2.13e64T^{2} \)
97 \( 1 - 4.84e32iT - 3.65e65T^{2} \)
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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

−11.88407585717827659476738825907, −9.763286711459023318367015771665, −8.672965953819221211431385927046, −7.57822188280177280552188596707, −6.75018016504118560731951358259, −6.39697606974394641979843596040, −4.73856239416812249182759773056, −3.84416119241801270133099970908, −1.49258555100176961134242711108, −0.62082918756665698136625786067, 0.61292131097700950164597956546, 1.84204087753400759778753592811, 2.87585005539811733181318198526, 3.89380041802622521165191720247, 4.56629432792764033702605081449, 5.81750708865092065861956507503, 8.574004083715004413833926071056, 9.126100293122948194051962237043, 10.28686112705090042727918921967, 10.82756889000433769920772541180

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