Properties

Label 2-75-15.8-c9-0-5
Degree $2$
Conductor $75$
Sign $0.979 - 0.200i$
Analytic cond. $38.6276$
Root an. cond. $6.21511$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−27.8 − 27.8i)2-s + (−131. − 48.2i)3-s + 1.03e3i·4-s + (2.32e3 + 5.00e3i)6-s + (6.30e3 − 6.30e3i)7-s + (1.45e4 − 1.45e4i)8-s + (1.50e4 + 1.27e4i)9-s − 1.00e4i·11-s + (5.00e4 − 1.36e5i)12-s + (−1.13e5 − 1.13e5i)13-s − 3.50e5·14-s − 2.80e5·16-s + (4.18e5 + 4.18e5i)17-s + (−6.41e4 − 7.71e5i)18-s − 7.43e4i·19-s + ⋯
L(s)  = 1  + (−1.22 − 1.22i)2-s + (−0.938 − 0.344i)3-s + 2.02i·4-s + (0.731 + 1.57i)6-s + (0.991 − 0.991i)7-s + (1.25 − 1.25i)8-s + (0.763 + 0.646i)9-s − 0.206i·11-s + (0.696 − 1.89i)12-s + (−1.10 − 1.10i)13-s − 2.43·14-s − 1.07·16-s + (1.21 + 1.21i)17-s + (−0.144 − 1.73i)18-s − 0.130i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.200i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.979 - 0.200i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.979 - 0.200i$
Analytic conductor: \(38.6276\)
Root analytic conductor: \(6.21511\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :9/2),\ 0.979 - 0.200i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.255548 + 0.0259435i\)
\(L(\frac12)\) \(\approx\) \(0.255548 + 0.0259435i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (131. + 48.2i)T \)
5 \( 1 \)
good2 \( 1 + (27.8 + 27.8i)T + 512iT^{2} \)
7 \( 1 + (-6.30e3 + 6.30e3i)T - 4.03e7iT^{2} \)
11 \( 1 + 1.00e4iT - 2.35e9T^{2} \)
13 \( 1 + (1.13e5 + 1.13e5i)T + 1.06e10iT^{2} \)
17 \( 1 + (-4.18e5 - 4.18e5i)T + 1.18e11iT^{2} \)
19 \( 1 + 7.43e4iT - 3.22e11T^{2} \)
23 \( 1 + (1.15e6 - 1.15e6i)T - 1.80e12iT^{2} \)
29 \( 1 + 5.61e6T + 1.45e13T^{2} \)
31 \( 1 + 1.72e6T + 2.64e13T^{2} \)
37 \( 1 + (-5.75e6 + 5.75e6i)T - 1.29e14iT^{2} \)
41 \( 1 - 1.54e7iT - 3.27e14T^{2} \)
43 \( 1 + (2.43e7 + 2.43e7i)T + 5.02e14iT^{2} \)
47 \( 1 + (8.82e6 + 8.82e6i)T + 1.11e15iT^{2} \)
53 \( 1 + (4.01e7 - 4.01e7i)T - 3.29e15iT^{2} \)
59 \( 1 - 1.04e8T + 8.66e15T^{2} \)
61 \( 1 - 1.51e7T + 1.16e16T^{2} \)
67 \( 1 + (5.07e7 - 5.07e7i)T - 2.72e16iT^{2} \)
71 \( 1 + 4.26e7iT - 4.58e16T^{2} \)
73 \( 1 + (-3.98e7 - 3.98e7i)T + 5.88e16iT^{2} \)
79 \( 1 - 1.98e8iT - 1.19e17T^{2} \)
83 \( 1 + (-1.82e7 + 1.82e7i)T - 1.86e17iT^{2} \)
89 \( 1 - 7.56e8T + 3.50e17T^{2} \)
97 \( 1 + (8.05e8 - 8.05e8i)T - 7.60e17iT^{2} \)
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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

−12.25735909175553468485832837994, −11.34416166592017599679866640308, −10.51801511118201419796461557021, −9.866960674679816570751119042630, −7.977605827231769560078135161456, −7.53845928693069929073574286392, −5.43790114306611053328217975232, −3.75565583071546426657663485729, −1.87167136854507473916630458684, −0.941008055204064641353037824142, 0.16420147883217560514570593755, 1.74519845206968323086732713241, 4.82638485803227785993640959500, 5.64984173012284654249708046945, 6.90657326386774839877528629519, 7.926803576888979888887823498027, 9.280685279191258862181268836120, 9.928926802268576897084654880958, 11.39667797890508446271089941283, 12.18923112563841298007774267591

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