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Properties

Label 2-75-15.2-c7-0-28
Degree $2$
Conductor $75$
Sign $0.466 + 0.884i$
Analytic cond. $23.4288$
Root an. cond. $4.84033$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.73 + 2.73i)2-s + (1.50 + 46.7i)3-s + 113. i·4-s + (−131. − 123. i)6-s + (−186. − 186. i)7-s + (−659. − 659. i)8-s + (−2.18e3 + 140. i)9-s − 6.62e3i·11-s + (−5.28e3 + 169. i)12-s + (−1.11e3 + 1.11e3i)13-s + 1.02e3·14-s − 1.08e4·16-s + (−8.36e3 + 8.36e3i)17-s + (5.58e3 − 6.35e3i)18-s − 2.40e4i·19-s + ⋯
L(s)  = 1  + (−0.241 + 0.241i)2-s + (0.0321 + 0.999i)3-s + 0.883i·4-s + (−0.249 − 0.233i)6-s + (−0.205 − 0.205i)7-s + (−0.455 − 0.455i)8-s + (−0.997 + 0.0641i)9-s − 1.50i·11-s + (−0.882 + 0.0283i)12-s + (−0.141 + 0.141i)13-s + 0.0993·14-s − 0.662·16-s + (−0.412 + 0.412i)17-s + (0.225 − 0.256i)18-s − 0.805i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.466 + 0.884i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.466 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.466 + 0.884i$
Analytic conductor: \(23.4288\)
Root analytic conductor: \(4.84033\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :7/2),\ 0.466 + 0.884i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.317559 - 0.191605i\)
\(L(\frac12)\) \(\approx\) \(0.317559 - 0.191605i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.50 - 46.7i)T \)
5 \( 1 \)
good2 \( 1 + (2.73 - 2.73i)T - 128iT^{2} \)
7 \( 1 + (186. + 186. i)T + 8.23e5iT^{2} \)
11 \( 1 + 6.62e3iT - 1.94e7T^{2} \)
13 \( 1 + (1.11e3 - 1.11e3i)T - 6.27e7iT^{2} \)
17 \( 1 + (8.36e3 - 8.36e3i)T - 4.10e8iT^{2} \)
19 \( 1 + 2.40e4iT - 8.93e8T^{2} \)
23 \( 1 + (-6.07e4 - 6.07e4i)T + 3.40e9iT^{2} \)
29 \( 1 - 7.63e4T + 1.72e10T^{2} \)
31 \( 1 + 3.09e5T + 2.75e10T^{2} \)
37 \( 1 + (2.94e4 + 2.94e4i)T + 9.49e10iT^{2} \)
41 \( 1 + 7.07e5iT - 1.94e11T^{2} \)
43 \( 1 + (-3.53e5 + 3.53e5i)T - 2.71e11iT^{2} \)
47 \( 1 + (5.32e5 - 5.32e5i)T - 5.06e11iT^{2} \)
53 \( 1 + (-8.10e5 - 8.10e5i)T + 1.17e12iT^{2} \)
59 \( 1 + 4.77e5T + 2.48e12T^{2} \)
61 \( 1 - 6.89e5T + 3.14e12T^{2} \)
67 \( 1 + (2.32e6 + 2.32e6i)T + 6.06e12iT^{2} \)
71 \( 1 - 3.82e6iT - 9.09e12T^{2} \)
73 \( 1 + (3.50e6 - 3.50e6i)T - 1.10e13iT^{2} \)
79 \( 1 + 6.25e5iT - 1.92e13T^{2} \)
83 \( 1 + (4.67e6 + 4.67e6i)T + 2.71e13iT^{2} \)
89 \( 1 + 4.99e6T + 4.42e13T^{2} \)
97 \( 1 + (3.38e6 + 3.38e6i)T + 8.07e13iT^{2} \)
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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

−13.01389164051582326620392711908, −11.54135764100298501354086489902, −10.76087854931962275837169113753, −9.190289805017866308187075775568, −8.608939147556287780159515126747, −7.14283003610230187042067799613, −5.59715398616284398152077077499, −3.96250833157793838918246059250, −2.99082876871103052234553136994, −0.13740935083256622429665377246, 1.39336394192128500644546158332, 2.53391532834178999503710005957, 4.93890243460331159088328601932, 6.29603178347841178817757481120, 7.33565431881646012287333771448, 8.833438158853428074246475981899, 9.898855141318129014004705423052, 11.11366102223098501308180785546, 12.26812767492301115055801828796, 13.10320104367828835883440162971

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