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Properties

Label 2-75-5.2-c8-0-16
Degree $2$
Conductor $75$
Sign $-0.326 + 0.945i$
Analytic cond. $30.5533$
Root an. cond. $5.52751$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.85 − 5.85i)2-s + (33.0 − 33.0i)3-s − 187. i·4-s − 387.·6-s + (1.10e3 + 1.10e3i)7-s + (−2.59e3 + 2.59e3i)8-s − 2.18e3i·9-s + 2.03e4·11-s + (−6.19e3 − 6.19e3i)12-s + (3.80e4 − 3.80e4i)13-s − 1.29e4i·14-s − 1.75e4·16-s + (3.50e4 + 3.50e4i)17-s + (−1.28e4 + 1.28e4i)18-s + 3.37e4i·19-s + ⋯
L(s)  = 1  + (−0.366 − 0.366i)2-s + (0.408 − 0.408i)3-s − 0.731i·4-s − 0.298·6-s + (0.460 + 0.460i)7-s + (−0.634 + 0.634i)8-s − 0.333i·9-s + 1.38·11-s + (−0.298 − 0.298i)12-s + (1.33 − 1.33i)13-s − 0.337i·14-s − 0.267·16-s + (0.420 + 0.420i)17-s + (−0.122 + 0.122i)18-s + 0.259i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.326 + 0.945i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.326 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-0.326 + 0.945i$
Analytic conductor: \(30.5533\)
Root analytic conductor: \(5.52751\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :4),\ -0.326 + 0.945i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.25459 - 1.76053i\)
\(L(\frac12)\) \(\approx\) \(1.25459 - 1.76053i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-33.0 + 33.0i)T \)
5 \( 1 \)
good2 \( 1 + (5.85 + 5.85i)T + 256iT^{2} \)
7 \( 1 + (-1.10e3 - 1.10e3i)T + 5.76e6iT^{2} \)
11 \( 1 - 2.03e4T + 2.14e8T^{2} \)
13 \( 1 + (-3.80e4 + 3.80e4i)T - 8.15e8iT^{2} \)
17 \( 1 + (-3.50e4 - 3.50e4i)T + 6.97e9iT^{2} \)
19 \( 1 - 3.37e4iT - 1.69e10T^{2} \)
23 \( 1 + (-1.80e5 + 1.80e5i)T - 7.83e10iT^{2} \)
29 \( 1 - 1.26e5iT - 5.00e11T^{2} \)
31 \( 1 + 6.06e5T + 8.52e11T^{2} \)
37 \( 1 + (3.25e4 + 3.25e4i)T + 3.51e12iT^{2} \)
41 \( 1 + 4.66e6T + 7.98e12T^{2} \)
43 \( 1 + (-2.52e6 + 2.52e6i)T - 1.16e13iT^{2} \)
47 \( 1 + (2.01e6 + 2.01e6i)T + 2.38e13iT^{2} \)
53 \( 1 + (-6.90e6 + 6.90e6i)T - 6.22e13iT^{2} \)
59 \( 1 + 3.09e6iT - 1.46e14T^{2} \)
61 \( 1 + 2.67e7T + 1.91e14T^{2} \)
67 \( 1 + (5.43e6 + 5.43e6i)T + 4.06e14iT^{2} \)
71 \( 1 - 4.05e7T + 6.45e14T^{2} \)
73 \( 1 + (1.17e6 - 1.17e6i)T - 8.06e14iT^{2} \)
79 \( 1 - 5.57e7iT - 1.51e15T^{2} \)
83 \( 1 + (-3.11e7 + 3.11e7i)T - 2.25e15iT^{2} \)
89 \( 1 - 5.37e7iT - 3.93e15T^{2} \)
97 \( 1 + (9.15e7 + 9.15e7i)T + 7.83e15iT^{2} \)
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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

−12.42259018536041880497989594738, −11.33990490870705233477219726495, −10.34135223731101704488154957060, −9.009532768751014274553582866304, −8.306561392438219864559285239106, −6.53555431350974582363863960803, −5.45026887324897837789457440392, −3.47229075199315083674563730964, −1.79762087129215797044033087395, −0.842691682606548751079827274909, 1.37736972321937291750233778672, 3.42966321377347033442159221715, 4.34163712261693833280714501477, 6.44206783738059629356515330574, 7.51013482845622888007722184717, 8.807252838057205523138229145208, 9.355268596511969991181453454484, 11.10413216673510332222273252720, 11.92925126984598653719849114632, 13.46667162546709749057292138700

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