Properties

Degree $2$
Conductor $25$
Sign $-0.894 - 0.447i$
Motivic weight $5$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.26i·2-s + 25.5i·3-s + 4.31·4-s − 134.·6-s − 131. i·7-s + 191. i·8-s − 408.·9-s + 290.·11-s + 110. i·12-s + 68.3i·13-s + 689.·14-s − 867.·16-s + 310. i·17-s − 2.14e3i·18-s + 2.13e3·19-s + ⋯
L(s)  = 1  + 0.930i·2-s + 1.63i·3-s + 0.134·4-s − 1.52·6-s − 1.01i·7-s + 1.05i·8-s − 1.68·9-s + 0.722·11-s + 0.220i·12-s + 0.112i·13-s + 0.940·14-s − 0.847·16-s + 0.260i·17-s − 1.56i·18-s + 1.35·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

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

Particular Values

\(L(3)\) \(\approx\) \(0.364966 + 1.54602i\)
\(L(\frac12)\) \(\approx\) \(0.364966 + 1.54602i\)
\(L(\frac{7}{2})\) 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 \( 1 - 5.26iT - 32T^{2} \)
3 \( 1 - 25.5iT - 243T^{2} \)
7 \( 1 + 131. iT - 1.68e4T^{2} \)
11 \( 1 - 290.T + 1.61e5T^{2} \)
13 \( 1 - 68.3iT - 3.71e5T^{2} \)
17 \( 1 - 310. iT - 1.41e6T^{2} \)
19 \( 1 - 2.13e3T + 2.47e6T^{2} \)
23 \( 1 - 873. iT - 6.43e6T^{2} \)
29 \( 1 - 2.58e3T + 2.05e7T^{2} \)
31 \( 1 + 9.08e3T + 2.86e7T^{2} \)
37 \( 1 + 3.99e3iT - 6.93e7T^{2} \)
41 \( 1 - 1.69e4T + 1.15e8T^{2} \)
43 \( 1 + 1.80e4iT - 1.47e8T^{2} \)
47 \( 1 + 2.48e4iT - 2.29e8T^{2} \)
53 \( 1 + 7.65e3iT - 4.18e8T^{2} \)
59 \( 1 - 9.23e3T + 7.14e8T^{2} \)
61 \( 1 - 3.32e3T + 8.44e8T^{2} \)
67 \( 1 - 3.23e4iT - 1.35e9T^{2} \)
71 \( 1 + 3.58e4T + 1.80e9T^{2} \)
73 \( 1 + 2.65e4iT - 2.07e9T^{2} \)
79 \( 1 + 7.17e4T + 3.07e9T^{2} \)
83 \( 1 - 3.96e4iT - 3.93e9T^{2} \)
89 \( 1 - 1.17e5T + 5.58e9T^{2} \)
97 \( 1 + 2.18e4iT - 8.58e9T^{2} \)
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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

−16.59711213503903963940664004456, −16.01006638096400380192490264835, −14.84813859249048005692900210131, −14.00304120436056502546286455342, −11.49928751644936514407336830291, −10.39122002952179453712342248818, −9.036303311022754685227716307362, −7.26605538204975956496636512817, −5.44693637640829885635126766000, −3.84125762573755899005461780097, 1.26538541216834812006344965203, 2.73982120197850173732765105466, 6.13054596981145812317308703548, 7.48680807645462244794859444809, 9.258705529028324016959758688327, 11.33436925882232191672499237582, 12.14540939484042289857060710099, 12.94390924950022308158447352168, 14.37482572224637157464159922416, 16.09079061051975252768894475171

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