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Properties

Label 2-75-5.4-c9-0-10
Degree $2$
Conductor $75$
Sign $-0.894 + 0.447i$
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  + 24.8i·2-s + 81i·3-s − 107.·4-s − 2.01e3·6-s + 4.01e3i·7-s + 1.00e4i·8-s − 6.56e3·9-s + 8.48e4·11-s − 8.68e3i·12-s + 1.19e5i·13-s − 9.97e4·14-s − 3.05e5·16-s − 1.16e5i·17-s − 1.63e5i·18-s + 2.34e5·19-s + ⋯
L(s)  = 1  + 1.09i·2-s + 0.577i·3-s − 0.209·4-s − 0.634·6-s + 0.631i·7-s + 0.869i·8-s − 0.333·9-s + 1.74·11-s − 0.120i·12-s + 1.15i·13-s − 0.694·14-s − 1.16·16-s − 0.339i·17-s − 0.366i·18-s + 0.413·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(38.6276\)
Root analytic conductor: \(6.21511\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :9/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.490775 - 2.07895i\)
\(L(\frac12)\) \(\approx\) \(0.490775 - 2.07895i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 81iT \)
5 \( 1 \)
good2 \( 1 - 24.8iT - 512T^{2} \)
7 \( 1 - 4.01e3iT - 4.03e7T^{2} \)
11 \( 1 - 8.48e4T + 2.35e9T^{2} \)
13 \( 1 - 1.19e5iT - 1.06e10T^{2} \)
17 \( 1 + 1.16e5iT - 1.18e11T^{2} \)
19 \( 1 - 2.34e5T + 3.22e11T^{2} \)
23 \( 1 - 2.34e6iT - 1.80e12T^{2} \)
29 \( 1 - 4.64e5T + 1.45e13T^{2} \)
31 \( 1 + 5.11e6T + 2.64e13T^{2} \)
37 \( 1 + 8.69e6iT - 1.29e14T^{2} \)
41 \( 1 + 9.05e6T + 3.27e14T^{2} \)
43 \( 1 - 8.63e6iT - 5.02e14T^{2} \)
47 \( 1 + 3.31e7iT - 1.11e15T^{2} \)
53 \( 1 + 6.41e7iT - 3.29e15T^{2} \)
59 \( 1 + 1.49e8T + 8.66e15T^{2} \)
61 \( 1 - 1.54e8T + 1.16e16T^{2} \)
67 \( 1 + 2.72e8iT - 2.72e16T^{2} \)
71 \( 1 + 3.56e8T + 4.58e16T^{2} \)
73 \( 1 - 2.06e8iT - 5.88e16T^{2} \)
79 \( 1 - 4.04e8T + 1.19e17T^{2} \)
83 \( 1 + 5.17e6iT - 1.86e17T^{2} \)
89 \( 1 + 4.32e8T + 3.50e17T^{2} \)
97 \( 1 - 1.32e9iT - 7.60e17T^{2} \)
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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

−13.84059498626701322401229238393, −11.87741697256381024803892842490, −11.35889759182844181879464035576, −9.442016796220128351019503700200, −8.861995564043158600550407689141, −7.31516435750804656014312994521, −6.29693539090020827415507835502, −5.18696901058967816579471242170, −3.73021942516890110223204600382, −1.81531133731932297085009611361, 0.62942316954173955887747653092, 1.50626965569955646578854619747, 3.00052107820511371635171061409, 4.17717680240127845938505553755, 6.23674699588393933795335065298, 7.26892500999050496264216908578, 8.797115523599284500629425949142, 10.10952212941846925590033572972, 11.02080428645665320254311601168, 12.07423921802445110562011329922

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