Properties

Label 2-75-75.8-c1-0-7
Degree $2$
Conductor $75$
Sign $-0.352 + 0.935i$
Analytic cond. $0.598878$
Root an. cond. $0.773872$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.12 − 2.20i)2-s + (−1.73 + 0.0162i)3-s + (−2.43 − 3.35i)4-s + (2.18 + 0.479i)5-s + (−1.91 + 3.84i)6-s + (−1.56 + 1.56i)7-s + (−5.24 + 0.829i)8-s + (2.99 − 0.0563i)9-s + (3.51 − 4.28i)10-s + (2.96 − 0.963i)11-s + (4.27 + 5.76i)12-s + (−1.61 + 0.821i)13-s + (1.69 + 5.21i)14-s + (−3.79 − 0.795i)15-s + (−1.50 + 4.62i)16-s + (0.290 + 1.83i)17-s + ⋯
L(s)  = 1  + (0.795 − 1.56i)2-s + (−0.999 + 0.00938i)3-s + (−1.21 − 1.67i)4-s + (0.976 + 0.214i)5-s + (−0.780 + 1.56i)6-s + (−0.591 + 0.591i)7-s + (−1.85 + 0.293i)8-s + (0.999 − 0.0187i)9-s + (1.11 − 1.35i)10-s + (0.894 − 0.290i)11-s + (1.23 + 1.66i)12-s + (−0.447 + 0.227i)13-s + (0.452 + 1.39i)14-s + (−0.978 − 0.205i)15-s + (−0.375 + 1.15i)16-s + (0.0704 + 0.444i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.352 + 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.352 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-0.352 + 0.935i$
Analytic conductor: \(0.598878\)
Root analytic conductor: \(0.773872\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :1/2),\ -0.352 + 0.935i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.610984 - 0.883187i\)
\(L(\frac12)\) \(\approx\) \(0.610984 - 0.883187i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.73 - 0.0162i)T \)
5 \( 1 + (-2.18 - 0.479i)T \)
good2 \( 1 + (-1.12 + 2.20i)T + (-1.17 - 1.61i)T^{2} \)
7 \( 1 + (1.56 - 1.56i)T - 7iT^{2} \)
11 \( 1 + (-2.96 + 0.963i)T + (8.89 - 6.46i)T^{2} \)
13 \( 1 + (1.61 - 0.821i)T + (7.64 - 10.5i)T^{2} \)
17 \( 1 + (-0.290 - 1.83i)T + (-16.1 + 5.25i)T^{2} \)
19 \( 1 + (3.62 - 4.98i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + (3.38 + 1.72i)T + (13.5 + 18.6i)T^{2} \)
29 \( 1 + (1.02 - 0.743i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (2.59 + 1.88i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.20 + 4.32i)T + (-21.7 + 29.9i)T^{2} \)
41 \( 1 + (3.49 + 1.13i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + (-0.478 - 0.478i)T + 43iT^{2} \)
47 \( 1 + (-11.8 - 1.87i)T + (44.6 + 14.5i)T^{2} \)
53 \( 1 + (-0.590 + 3.72i)T + (-50.4 - 16.3i)T^{2} \)
59 \( 1 + (-3.54 + 10.9i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (1.39 + 4.29i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (2.12 - 0.336i)T + (63.7 - 20.7i)T^{2} \)
71 \( 1 + (-2.82 - 3.88i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.61 + 5.12i)T + (-42.9 - 59.0i)T^{2} \)
79 \( 1 + (0.567 + 0.781i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (-0.118 + 0.0187i)T + (78.9 - 25.6i)T^{2} \)
89 \( 1 + (-3.39 - 10.4i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (2.60 - 16.4i)T + (-92.2 - 29.9i)T^{2} \)
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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

−13.87613408763859118203460048459, −12.63660523757282963052275855258, −12.24991299942001261679613149757, −11.01741877592389462892926355710, −10.15431157482347117407309935058, −9.294439473222933523313914269656, −6.37147267531479979525556642862, −5.51817774144503337816796216636, −3.92240161138485096779805130610, −1.96099295766294966481465738939, 4.28069564635971403708117809199, 5.44171916452497510359779184785, 6.50603228901584779784797492327, 7.18786055719461395212791943303, 9.093533717492865705853717212366, 10.32049731107730023845982517785, 12.08593267393655793027384534460, 13.10949346485810731135416315574, 13.77402963046402860644081222001, 14.97575901658941623434902831430

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