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  − 1.26e5i·2-s + 8.63e7i·3-s − 7.38e9·4-s + 1.09e13·6-s + 7.78e13i·7-s − 1.52e14i·8-s − 1.89e15·9-s + 1.61e17·11-s − 6.37e17i·12-s + 2.28e18i·13-s + 9.84e18·14-s − 8.26e19·16-s + 2.46e20i·17-s + 2.39e20i·18-s + 1.53e21·19-s + ⋯
L(s)  = 1  − 1.36i·2-s + 1.15i·3-s − 0.859·4-s + 1.57·6-s + 0.885i·7-s − 0.190i·8-s − 0.340·9-s + 1.06·11-s − 0.995i·12-s + 0.953i·13-s + 1.20·14-s − 1.12·16-s + 1.22i·17-s + 0.464i·18-s + 1.22·19-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

\(L(17)\) \(\approx\) \(2.421271244\)
\(L(\frac12)\) \(\approx\) \(2.421271244\)
\(L(\frac{35}{2})\) 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 \( 1 + 1.26e5iT - 8.58e9T^{2} \)
3 \( 1 - 8.63e7iT - 5.55e15T^{2} \)
7 \( 1 - 7.78e13iT - 7.73e27T^{2} \)
11 \( 1 - 1.61e17T + 2.32e34T^{2} \)
13 \( 1 - 2.28e18iT - 5.75e36T^{2} \)
17 \( 1 - 2.46e20iT - 4.02e40T^{2} \)
19 \( 1 - 1.53e21T + 1.58e42T^{2} \)
23 \( 1 - 2.16e22iT - 8.65e44T^{2} \)
29 \( 1 - 2.07e23T + 1.81e48T^{2} \)
31 \( 1 - 1.75e24T + 1.64e49T^{2} \)
37 \( 1 - 8.12e25iT - 5.63e51T^{2} \)
41 \( 1 + 1.92e26T + 1.66e53T^{2} \)
43 \( 1 + 8.01e26iT - 8.02e53T^{2} \)
47 \( 1 + 7.53e27iT - 1.51e55T^{2} \)
53 \( 1 + 1.75e28iT - 7.96e56T^{2} \)
59 \( 1 - 2.97e29T + 2.74e58T^{2} \)
61 \( 1 + 3.70e29T + 8.23e58T^{2} \)
67 \( 1 - 8.43e29iT - 1.82e60T^{2} \)
71 \( 1 - 5.92e30T + 1.23e61T^{2} \)
73 \( 1 + 7.49e30iT - 3.08e61T^{2} \)
79 \( 1 + 6.07e30T + 4.18e62T^{2} \)
83 \( 1 + 3.47e31iT - 2.13e63T^{2} \)
89 \( 1 + 1.80e32T + 2.13e64T^{2} \)
97 \( 1 - 1.09e33iT - 3.65e65T^{2} \)
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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

−11.57244629782409086430162114320, −10.31820355141290402889945157595, −9.525496509577793694579678921934, −8.765720903277811405221571352071, −6.68529454759073505808639205017, −5.19269208926660243135698163012, −3.98709648103495369937593367197, −3.43301629114237665417918929108, −2.06667259525978232056771891243, −1.19812121222543098061826626943, 0.50016107173652805996046967018, 1.22450504257602700691331416713, 2.76865270434688444327370234241, 4.40336730566162496818589336163, 5.69672952262498442218788803440, 6.78469073980243290122171354569, 7.31897477792134968848631340029, 8.161048427139709221348171252909, 9.583078690694664387238123455100, 11.27000181051433100829356760578

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