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Properties

Label 2-75-5.4-c11-0-2
Degree $2$
Conductor $75$
Sign $-0.894 + 0.447i$
Analytic cond. $57.6257$
Root an. cond. $7.59116$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 27.7i·2-s − 243i·3-s + 1.27e3·4-s + 6.73e3·6-s + 7.21e4i·7-s + 9.22e4i·8-s − 5.90e4·9-s − 5.09e5·11-s − 3.10e5i·12-s + 1.85e6i·13-s − 2.00e6·14-s + 6.39e4·16-s − 5.94e6i·17-s − 1.63e6i·18-s − 6.02e6·19-s + ⋯
L(s)  = 1  + 0.612i·2-s − 0.577i·3-s + 0.624·4-s + 0.353·6-s + 1.62i·7-s + 0.995i·8-s − 0.333·9-s − 0.953·11-s − 0.360i·12-s + 1.38i·13-s − 0.993·14-s + 0.0152·16-s − 1.01i·17-s − 0.204i·18-s − 0.557·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(57.6257\)
Root analytic conductor: \(7.59116\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :11/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.159741 - 0.676677i\)
\(L(\frac12)\) \(\approx\) \(0.159741 - 0.676677i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 243iT \)
5 \( 1 \)
good2 \( 1 - 27.7iT - 2.04e3T^{2} \)
7 \( 1 - 7.21e4iT - 1.97e9T^{2} \)
11 \( 1 + 5.09e5T + 2.85e11T^{2} \)
13 \( 1 - 1.85e6iT - 1.79e12T^{2} \)
17 \( 1 + 5.94e6iT - 3.42e13T^{2} \)
19 \( 1 + 6.02e6T + 1.16e14T^{2} \)
23 \( 1 + 4.82e7iT - 9.52e14T^{2} \)
29 \( 1 - 1.13e7T + 1.22e16T^{2} \)
31 \( 1 + 1.72e8T + 2.54e16T^{2} \)
37 \( 1 + 6.25e8iT - 1.77e17T^{2} \)
41 \( 1 + 5.53e8T + 5.50e17T^{2} \)
43 \( 1 - 1.52e9iT - 9.29e17T^{2} \)
47 \( 1 + 1.19e9iT - 2.47e18T^{2} \)
53 \( 1 - 1.22e9iT - 9.26e18T^{2} \)
59 \( 1 - 5.83e9T + 3.01e19T^{2} \)
61 \( 1 + 6.61e9T + 4.35e19T^{2} \)
67 \( 1 + 1.66e10iT - 1.22e20T^{2} \)
71 \( 1 - 7.36e9T + 2.31e20T^{2} \)
73 \( 1 + 6.35e9iT - 3.13e20T^{2} \)
79 \( 1 + 2.47e10T + 7.47e20T^{2} \)
83 \( 1 - 3.59e10iT - 1.28e21T^{2} \)
89 \( 1 + 7.47e10T + 2.77e21T^{2} \)
97 \( 1 - 1.66e10iT - 7.15e21T^{2} \)
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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

−12.70626528863705437016431563065, −11.89550317603326377200749123891, −10.96654209549283072859822614483, −9.161640686348273698541328095454, −8.241929340180591801774147832486, −7.01409652882461435616368042707, −6.07992754225463552015366862378, −5.03036382819089020418351893887, −2.62564721354921405290832439307, −2.01993023831638046132753960468, 0.15624714658301114180173171977, 1.46282877435857133469932798388, 3.09569407100046179688279306752, 3.96221341898186269708131758379, 5.55793132076411557049227913736, 7.10602869238327033259703459156, 8.090156323438697338575153740832, 10.07665452023281019132720100977, 10.43163412388642374460715923778, 11.26287383108580357455636067813

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