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Properties

Label 2-75-5.4-c17-0-4
Degree $2$
Conductor $75$
Sign $0.894 - 0.447i$
Analytic cond. $137.416$
Root an. cond. $11.7224$
Motivic weight $17$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 257. i·2-s − 6.56e3i·3-s + 6.47e4·4-s − 1.68e6·6-s − 8.43e6i·7-s − 5.04e7i·8-s − 4.30e7·9-s − 1.27e9·11-s − 4.24e8i·12-s + 2.16e9i·13-s − 2.17e9·14-s − 4.50e9·16-s − 3.94e10i·17-s + 1.10e10i·18-s − 1.12e11·19-s + ⋯
L(s)  = 1  − 0.711i·2-s − 0.577i·3-s + 0.493·4-s − 0.410·6-s − 0.553i·7-s − 1.06i·8-s − 0.333·9-s − 1.79·11-s − 0.285i·12-s + 0.737i·13-s − 0.393·14-s − 0.262·16-s − 1.37i·17-s + 0.237i·18-s − 1.51·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(137.416\)
Root analytic conductor: \(11.7224\)
Motivic weight: \(17\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :17/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(9)\) \(\approx\) \(0.5553607614\)
\(L(\frac12)\) \(\approx\) \(0.5553607614\)
\(L(\frac{19}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 6.56e3iT \)
5 \( 1 \)
good2 \( 1 + 257. iT - 1.31e5T^{2} \)
7 \( 1 + 8.43e6iT - 2.32e14T^{2} \)
11 \( 1 + 1.27e9T + 5.05e17T^{2} \)
13 \( 1 - 2.16e9iT - 8.65e18T^{2} \)
17 \( 1 + 3.94e10iT - 8.27e20T^{2} \)
19 \( 1 + 1.12e11T + 5.48e21T^{2} \)
23 \( 1 - 5.33e11iT - 1.41e23T^{2} \)
29 \( 1 - 1.70e12T + 7.25e24T^{2} \)
31 \( 1 + 1.16e11T + 2.25e25T^{2} \)
37 \( 1 + 3.69e13iT - 4.56e26T^{2} \)
41 \( 1 - 6.01e13T + 2.61e27T^{2} \)
43 \( 1 + 3.10e13iT - 5.87e27T^{2} \)
47 \( 1 - 2.34e14iT - 2.66e28T^{2} \)
53 \( 1 - 5.42e14iT - 2.05e29T^{2} \)
59 \( 1 - 1.85e14T + 1.27e30T^{2} \)
61 \( 1 + 1.71e15T + 2.24e30T^{2} \)
67 \( 1 - 6.75e12iT - 1.10e31T^{2} \)
71 \( 1 - 9.21e13T + 2.96e31T^{2} \)
73 \( 1 - 1.14e16iT - 4.74e31T^{2} \)
79 \( 1 + 8.98e15T + 1.81e32T^{2} \)
83 \( 1 - 4.25e15iT - 4.21e32T^{2} \)
89 \( 1 - 3.54e16T + 1.37e33T^{2} \)
97 \( 1 + 3.08e16iT - 5.95e33T^{2} \)
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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

−11.17919864173396275796312513156, −10.55796347741177998517382435460, −9.315386148581539852385875315801, −7.71330187009523997430142798061, −7.07601741510781359202498018421, −5.75218081800232588085727888413, −4.27663120037426391180255107793, −2.83301300654845254938276893404, −2.15325059367750834939771314179, −0.913195481312878490617303939243, 0.11420658652031265032896330553, 2.12975693701248716540828702601, 2.91966268045871198943036286674, 4.62886332256976230919323778552, 5.63865253550533935842834007635, 6.48390285559799391115177218001, 8.081845975599203910877979204036, 8.452253068007114987413000652920, 10.41469698908697595385544430910, 10.69400693238580110315243093453

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