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Properties

Label 2-75-3.2-c8-0-13
Degree $2$
Conductor $75$
Sign $0.488 + 0.872i$
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  − 24.1i·2-s + (−39.5 − 70.6i)3-s − 326.·4-s + (−1.70e3 + 955. i)6-s + 1.04e3·7-s + 1.70e3i·8-s + (−3.42e3 + 5.59e3i)9-s + 1.95e4i·11-s + (1.29e4 + 2.30e4i)12-s + 2.90e4·13-s − 2.51e4i·14-s − 4.24e4·16-s + 1.22e5i·17-s + (1.35e5 + 8.27e4i)18-s + 1.89e5·19-s + ⋯
L(s)  = 1  − 1.50i·2-s + (−0.488 − 0.872i)3-s − 1.27·4-s + (−1.31 + 0.737i)6-s + 0.434·7-s + 0.415i·8-s + (−0.522 + 0.852i)9-s + 1.33i·11-s + (0.623 + 1.11i)12-s + 1.01·13-s − 0.654i·14-s − 0.648·16-s + 1.46i·17-s + (1.28 + 0.788i)18-s + 1.45·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.488 + 0.872i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.488 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.488 + 0.872i$
Analytic conductor: \(30.5533\)
Root analytic conductor: \(5.52751\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :4),\ 0.488 + 0.872i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.14750 - 0.672602i\)
\(L(\frac12)\) \(\approx\) \(1.14750 - 0.672602i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (39.5 + 70.6i)T \)
5 \( 1 \)
good2 \( 1 + 24.1iT - 256T^{2} \)
7 \( 1 - 1.04e3T + 5.76e6T^{2} \)
11 \( 1 - 1.95e4iT - 2.14e8T^{2} \)
13 \( 1 - 2.90e4T + 8.15e8T^{2} \)
17 \( 1 - 1.22e5iT - 6.97e9T^{2} \)
19 \( 1 - 1.89e5T + 1.69e10T^{2} \)
23 \( 1 - 1.12e5iT - 7.83e10T^{2} \)
29 \( 1 - 1.08e5iT - 5.00e11T^{2} \)
31 \( 1 + 1.19e6T + 8.52e11T^{2} \)
37 \( 1 + 2.84e6T + 3.51e12T^{2} \)
41 \( 1 - 3.90e6iT - 7.98e12T^{2} \)
43 \( 1 + 8.64e5T + 1.16e13T^{2} \)
47 \( 1 - 1.48e6iT - 2.38e13T^{2} \)
53 \( 1 + 3.65e6iT - 6.22e13T^{2} \)
59 \( 1 + 1.46e7iT - 1.46e14T^{2} \)
61 \( 1 - 1.60e7T + 1.91e14T^{2} \)
67 \( 1 - 2.14e7T + 4.06e14T^{2} \)
71 \( 1 - 4.10e7iT - 6.45e14T^{2} \)
73 \( 1 - 4.70e7T + 8.06e14T^{2} \)
79 \( 1 + 2.43e7T + 1.51e15T^{2} \)
83 \( 1 - 6.88e6iT - 2.25e15T^{2} \)
89 \( 1 - 3.39e7iT - 3.93e15T^{2} \)
97 \( 1 - 1.50e7T + 7.83e15T^{2} \)
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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

−12.55681838685694940076478238134, −11.58648529646895321147777276014, −10.84008434360460877039845350307, −9.698956386977973047426915676797, −8.202307951936961842276828114143, −6.83093271162332445633774661537, −5.19120730662776510594138473139, −3.62139701994354469031339335486, −1.91892513911487202599344174671, −1.25243919009494480060137379185, 0.49689369784829477081516908870, 3.49144950422100285509492603347, 5.10359156188313264925770839475, 5.76211180845401700879050243945, 7.06051997681725820768628257583, 8.451171374118996923475637376760, 9.274341501470351133114127005627, 10.89529387102746679969853937732, 11.70429736774163975313750038976, 13.73725006157175225756069727627

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