Properties

Label 2-75-15.2-c9-0-17
Degree $2$
Conductor $75$
Sign $-0.229 - 0.973i$
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  + (11.0 − 11.0i)2-s + (99.2 + 99.2i)3-s + 269i·4-s + 2.18e3·6-s + (8.60e3 + 8.60e3i)8-s + 1.96e4i·9-s + (−2.66e4 + 2.66e4i)12-s + 5.20e4·16-s + (2.10e5 − 2.10e5i)17-s + (2.16e5 + 2.16e5i)18-s + 1.03e6i·19-s + (−3.47e5 − 3.47e5i)23-s + 1.70e6i·24-s + (−1.95e6 + 1.95e6i)27-s − 8.24e6·31-s + (−3.83e6 + 3.83e6i)32-s + ⋯
L(s)  = 1  + (0.487 − 0.487i)2-s + (0.707 + 0.707i)3-s + 0.525i·4-s + 0.688·6-s + (0.743 + 0.743i)8-s + 0.999i·9-s + (−0.371 + 0.371i)12-s + 0.198·16-s + (0.610 − 0.610i)17-s + (0.487 + 0.487i)18-s + 1.82i·19-s + (−0.258 − 0.258i)23-s + 1.05i·24-s + (−0.707 + 0.707i)27-s − 1.60·31-s + (−0.646 + 0.646i)32-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-0.229 - 0.973i$
Analytic conductor: \(38.6276\)
Root analytic conductor: \(6.21511\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :9/2),\ -0.229 - 0.973i)\)

Particular Values

\(L(5)\) \(\approx\) \(1.91432 + 2.41885i\)
\(L(\frac12)\) \(\approx\) \(1.91432 + 2.41885i\)
\(L(\frac{11}{2})\) 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 \( 1 + (-11.0 + 11.0i)T - 512iT^{2} \)
7 \( 1 + 4.03e7iT^{2} \)
11 \( 1 - 2.35e9T^{2} \)
13 \( 1 - 1.06e10iT^{2} \)
17 \( 1 + (-2.10e5 + 2.10e5i)T - 1.18e11iT^{2} \)
19 \( 1 - 1.03e6iT - 3.22e11T^{2} \)
23 \( 1 + (3.47e5 + 3.47e5i)T + 1.80e12iT^{2} \)
29 \( 1 + 1.45e13T^{2} \)
31 \( 1 + 8.24e6T + 2.64e13T^{2} \)
37 \( 1 + 1.29e14iT^{2} \)
41 \( 1 - 3.27e14T^{2} \)
43 \( 1 - 5.02e14iT^{2} \)
47 \( 1 + (4.70e7 - 4.70e7i)T - 1.11e15iT^{2} \)
53 \( 1 + (-7.68e7 - 7.68e7i)T + 3.29e15iT^{2} \)
59 \( 1 + 8.66e15T^{2} \)
61 \( 1 - 1.97e8T + 1.16e16T^{2} \)
67 \( 1 + 2.72e16iT^{2} \)
71 \( 1 - 4.58e16T^{2} \)
73 \( 1 - 5.88e16iT^{2} \)
79 \( 1 + 4.21e8iT - 1.19e17T^{2} \)
83 \( 1 + (-6.04e8 - 6.04e8i)T + 1.86e17iT^{2} \)
89 \( 1 + 3.50e17T^{2} \)
97 \( 1 + 7.60e17iT^{2} \)
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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

−12.98781878358313388913909100390, −11.97451118838611081602980676068, −10.79633032121567233098780722353, −9.725174445285090783898358937924, −8.416221566442216637647382441818, −7.51643527163353402367013206116, −5.40524694329017291118976640573, −4.08385392450591470051968928345, −3.21184237952587834073663444436, −1.90218631780234062999349876865, 0.66846494216827235557483573569, 2.01649066548814756855308894765, 3.67934268839615054305622414550, 5.25150332569875340860528679875, 6.54090218423603259375700624490, 7.43158229333381916323603782751, 8.813007274261545294510572573535, 9.952215117657774180479846014440, 11.36664014019512238863565212048, 12.83742803767994314173167080693

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