Properties

Label 2-75-15.8-c9-0-3
Degree $2$
Conductor $75$
Sign $-0.973 - 0.229i$
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  + (99.2 + 99.2i)3-s − 512i·4-s + (−1.25e3 + 1.25e3i)7-s + 1.96e4i·9-s + (5.07e4 − 5.07e4i)12-s + (−1.19e5 − 1.19e5i)13-s − 2.62e5·16-s + 9.76e5i·19-s − 2.49e5·21-s + (−1.95e6 + 1.95e6i)27-s + (6.43e5 + 6.43e5i)28-s − 1.69e6·31-s + 1.00e7·36-s + (−1.18e7 + 1.18e7i)37-s − 2.36e7i·39-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)3-s − i·4-s + (−0.197 + 0.197i)7-s + 0.999i·9-s + (0.707 − 0.707i)12-s + (−1.15 − 1.15i)13-s − 16-s + 1.71i·19-s − 0.279·21-s + (−0.707 + 0.707i)27-s + (0.197 + 0.197i)28-s − 0.328·31-s + 0.999·36-s + (−1.04 + 1.04i)37-s − 1.63i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.229i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-0.973 - 0.229i$
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: \((2,\ 75,\ (\ :9/2),\ -0.973 - 0.229i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.0458977 + 0.394196i\)
\(L(\frac12)\) \(\approx\) \(0.0458977 + 0.394196i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-99.2 - 99.2i)T \)
5 \( 1 \)
good2 \( 1 + 512iT^{2} \)
7 \( 1 + (1.25e3 - 1.25e3i)T - 4.03e7iT^{2} \)
11 \( 1 - 2.35e9T^{2} \)
13 \( 1 + (1.19e5 + 1.19e5i)T + 1.06e10iT^{2} \)
17 \( 1 + 1.18e11iT^{2} \)
19 \( 1 - 9.76e5iT - 3.22e11T^{2} \)
23 \( 1 - 1.80e12iT^{2} \)
29 \( 1 + 1.45e13T^{2} \)
31 \( 1 + 1.69e6T + 2.64e13T^{2} \)
37 \( 1 + (1.18e7 - 1.18e7i)T - 1.29e14iT^{2} \)
41 \( 1 - 3.27e14T^{2} \)
43 \( 1 + (2.94e7 + 2.94e7i)T + 5.02e14iT^{2} \)
47 \( 1 + 1.11e15iT^{2} \)
53 \( 1 - 3.29e15iT^{2} \)
59 \( 1 + 8.66e15T^{2} \)
61 \( 1 + 1.17e8T + 1.16e16T^{2} \)
67 \( 1 + (2.19e8 - 2.19e8i)T - 2.72e16iT^{2} \)
71 \( 1 - 4.58e16T^{2} \)
73 \( 1 + (2.71e8 + 2.71e8i)T + 5.88e16iT^{2} \)
79 \( 1 + 6.16e8iT - 1.19e17T^{2} \)
83 \( 1 - 1.86e17iT^{2} \)
89 \( 1 + 3.50e17T^{2} \)
97 \( 1 + (-8.30e8 + 8.30e8i)T - 7.60e17iT^{2} \)
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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

−13.42476741391544213615832673011, −12.05895267754208550213543936373, −10.38393752877010969381653520813, −10.07923198596928118167870296213, −8.837320862005306315340221882868, −7.56298565681549986283982865595, −5.81549055782276979553948310868, −4.80348974368373954967817238876, −3.20751323349692611058195441930, −1.79507032768832124632145415993, 0.093844874986505801546363170258, 2.05002086155482247184820147514, 3.14578044074516842061387578903, 4.53191575359585932695404139155, 6.78567365751891155390584756704, 7.37245134327516390497639798200, 8.675046737623734136584123105404, 9.508818482861755860764298602181, 11.40829267787463308060616683721, 12.31634297752545486394480761816

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