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Properties

Label 2-75-3.2-c8-0-19
Degree $2$
Conductor $75$
Sign $0.513 - 0.858i$
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  + 29.5i·2-s + (−41.5 + 69.5i)3-s − 614.·4-s + (−2.05e3 − 1.22e3i)6-s − 3.17e3·7-s − 1.05e4i·8-s + (−3.10e3 − 5.78e3i)9-s + 1.29e4i·11-s + (2.55e4 − 4.27e4i)12-s − 8.75e3·13-s − 9.36e4i·14-s + 1.54e5·16-s + 1.08e5i·17-s + (1.70e5 − 9.15e4i)18-s − 7.86e4·19-s + ⋯
L(s)  = 1  + 1.84i·2-s + (−0.513 + 0.858i)3-s − 2.39·4-s + (−1.58 − 0.946i)6-s − 1.32·7-s − 2.58i·8-s + (−0.472 − 0.881i)9-s + 0.887i·11-s + (1.23 − 2.05i)12-s − 0.306·13-s − 2.43i·14-s + 2.35·16-s + 1.30i·17-s + (1.62 − 0.872i)18-s − 0.603·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.513 - 0.858i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.513 - 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.513 - 0.858i$
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: \((2,\ 75,\ (\ :4),\ 0.513 - 0.858i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.0765561 + 0.0434128i\)
\(L(\frac12)\) \(\approx\) \(0.0765561 + 0.0434128i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (41.5 - 69.5i)T \)
5 \( 1 \)
good2 \( 1 - 29.5iT - 256T^{2} \)
7 \( 1 + 3.17e3T + 5.76e6T^{2} \)
11 \( 1 - 1.29e4iT - 2.14e8T^{2} \)
13 \( 1 + 8.75e3T + 8.15e8T^{2} \)
17 \( 1 - 1.08e5iT - 6.97e9T^{2} \)
19 \( 1 + 7.86e4T + 1.69e10T^{2} \)
23 \( 1 - 4.35e4iT - 7.83e10T^{2} \)
29 \( 1 + 1.83e5iT - 5.00e11T^{2} \)
31 \( 1 - 7.80e5T + 8.52e11T^{2} \)
37 \( 1 + 2.20e6T + 3.51e12T^{2} \)
41 \( 1 - 3.04e6iT - 7.98e12T^{2} \)
43 \( 1 + 4.84e6T + 1.16e13T^{2} \)
47 \( 1 + 3.51e6iT - 2.38e13T^{2} \)
53 \( 1 - 8.64e6iT - 6.22e13T^{2} \)
59 \( 1 + 5.16e6iT - 1.46e14T^{2} \)
61 \( 1 - 5.78e6T + 1.91e14T^{2} \)
67 \( 1 - 3.67e7T + 4.06e14T^{2} \)
71 \( 1 - 9.71e5iT - 6.45e14T^{2} \)
73 \( 1 + 9.46e6T + 8.06e14T^{2} \)
79 \( 1 - 5.63e7T + 1.51e15T^{2} \)
83 \( 1 + 5.88e7iT - 2.25e15T^{2} \)
89 \( 1 + 1.92e7iT - 3.93e15T^{2} \)
97 \( 1 - 1.42e8T + 7.83e15T^{2} \)
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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

−13.29962813765481980706991079773, −12.32754158023116871859251441380, −10.24433715253252793669171149896, −9.540187593177200865524331183618, −8.377725129300418801877352939907, −6.80821007602025401269306554332, −6.13630929625762953871345076401, −4.87866228977776776272677155842, −3.73196478178487610543072330031, −0.04522573540101939348734107839, 0.75602380708347498029748471825, 2.35931070172291969755720450661, 3.39099327224645045318490455421, 5.16095659739883390781592896712, 6.69220062564719809481071203126, 8.512833463414577005320951556366, 9.695189727682779971506853356073, 10.71846214000301245073473432269, 11.72282481012953943134530198286, 12.48515486698750583032161259669

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