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Properties

Label 2-75-15.2-c9-0-14
Degree $2$
Conductor $75$
Sign $0.0329 - 0.999i$
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  + (−15.0 + 15.0i)2-s + (−65.5 + 124. i)3-s + 60.0i·4-s + (−880. − 2.84e3i)6-s + (−4.27e3 − 4.27e3i)7-s + (−8.59e3 − 8.59e3i)8-s + (−1.10e4 − 1.62e4i)9-s + 5.87e4i·11-s + (−7.44e3 − 3.93e3i)12-s + (3.79e4 − 3.79e4i)13-s + 1.28e5·14-s + 2.27e5·16-s + (3.16e5 − 3.16e5i)17-s + (4.11e5 + 7.75e4i)18-s + 3.65e5i·19-s + ⋯
L(s)  = 1  + (−0.664 + 0.664i)2-s + (−0.466 + 0.884i)3-s + 0.117i·4-s + (−0.277 − 0.897i)6-s + (−0.673 − 0.673i)7-s + (−0.742 − 0.742i)8-s + (−0.563 − 0.825i)9-s + 1.20i·11-s + (−0.103 − 0.0547i)12-s + (0.368 − 0.368i)13-s + 0.894·14-s + 0.869·16-s + (0.919 − 0.919i)17-s + (0.923 + 0.174i)18-s + 0.643i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0329 - 0.999i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.0329 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.0329 - 0.999i$
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: \((2,\ 75,\ (\ :9/2),\ 0.0329 - 0.999i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.563178 + 0.544922i\)
\(L(\frac12)\) \(\approx\) \(0.563178 + 0.544922i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (65.5 - 124. i)T \)
5 \( 1 \)
good2 \( 1 + (15.0 - 15.0i)T - 512iT^{2} \)
7 \( 1 + (4.27e3 + 4.27e3i)T + 4.03e7iT^{2} \)
11 \( 1 - 5.87e4iT - 2.35e9T^{2} \)
13 \( 1 + (-3.79e4 + 3.79e4i)T - 1.06e10iT^{2} \)
17 \( 1 + (-3.16e5 + 3.16e5i)T - 1.18e11iT^{2} \)
19 \( 1 - 3.65e5iT - 3.22e11T^{2} \)
23 \( 1 + (1.50e6 + 1.50e6i)T + 1.80e12iT^{2} \)
29 \( 1 - 1.94e6T + 1.45e13T^{2} \)
31 \( 1 - 9.54e5T + 2.64e13T^{2} \)
37 \( 1 + (-8.41e6 - 8.41e6i)T + 1.29e14iT^{2} \)
41 \( 1 + 1.47e7iT - 3.27e14T^{2} \)
43 \( 1 + (2.54e7 - 2.54e7i)T - 5.02e14iT^{2} \)
47 \( 1 + (-5.21e6 + 5.21e6i)T - 1.11e15iT^{2} \)
53 \( 1 + (-8.10e6 - 8.10e6i)T + 3.29e15iT^{2} \)
59 \( 1 + 1.67e8T + 8.66e15T^{2} \)
61 \( 1 - 2.01e8T + 1.16e16T^{2} \)
67 \( 1 + (2.48e7 + 2.48e7i)T + 2.72e16iT^{2} \)
71 \( 1 - 3.74e7iT - 4.58e16T^{2} \)
73 \( 1 + (2.74e8 - 2.74e8i)T - 5.88e16iT^{2} \)
79 \( 1 - 2.73e8iT - 1.19e17T^{2} \)
83 \( 1 + (4.89e8 + 4.89e8i)T + 1.86e17iT^{2} \)
89 \( 1 - 4.96e8T + 3.50e17T^{2} \)
97 \( 1 + (-2.56e8 - 2.56e8i)T + 7.60e17iT^{2} \)
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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

−12.72808322984151850281243277776, −11.84361600814146735240619266332, −10.08070863402957718907955780749, −9.845678539832514350855687629466, −8.369824967148027911650938781485, −7.13734832799711249741499705850, −6.06802415013792434764487802103, −4.42089843727982605606422098872, −3.21177302761789918056716397155, −0.56697582361847542773914766330, 0.61276976130487292497056856342, 1.78123210342745889511567553416, 3.12435047228616585867592497752, 5.64382848637329399512949666131, 6.25336051275255368244401396078, 8.045937287912392402428331532118, 9.028195289152715735018784697857, 10.28762880664766655017485559709, 11.37666599655579274362919320633, 12.06632950076682022493112448289

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