Properties

Label 2-75-3.2-c2-0-6
Degree $2$
Conductor $75$
Sign $1$
Analytic cond. $2.04360$
Root an. cond. $1.42954$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 4·4-s − 11·7-s + 9·9-s + 12·12-s + 13-s + 16·16-s − 37·19-s − 33·21-s + 27·27-s − 44·28-s − 13·31-s + 36·36-s − 26·37-s + 3·39-s + 61·43-s + 48·48-s + 72·49-s + 4·52-s − 111·57-s + 47·61-s − 99·63-s + 64·64-s + 109·67-s + 46·73-s − 148·76-s − 142·79-s + ⋯
L(s)  = 1  + 3-s + 4-s − 1.57·7-s + 9-s + 12-s + 1/13·13-s + 16-s − 1.94·19-s − 1.57·21-s + 27-s − 1.57·28-s − 0.419·31-s + 36-s − 0.702·37-s + 1/13·39-s + 1.41·43-s + 48-s + 1.46·49-s + 1/13·52-s − 1.94·57-s + 0.770·61-s − 1.57·63-s + 64-s + 1.62·67-s + 0.630·73-s − 1.94·76-s − 1.79·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(2.04360\)
Root analytic conductor: \(1.42954\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: $\chi_{75} (26, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.752314020\)
\(L(\frac12)\) \(\approx\) \(1.752314020\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - p T \)
5 \( 1 \)
good2 \( ( 1 - p T )( 1 + p T ) \)
7 \( 1 + 11 T + p^{2} T^{2} \)
11 \( ( 1 - p T )( 1 + p T ) \)
13 \( 1 - T + p^{2} T^{2} \)
17 \( ( 1 - p T )( 1 + p T ) \)
19 \( 1 + 37 T + p^{2} T^{2} \)
23 \( ( 1 - p T )( 1 + p T ) \)
29 \( ( 1 - p T )( 1 + p T ) \)
31 \( 1 + 13 T + p^{2} T^{2} \)
37 \( 1 + 26 T + p^{2} T^{2} \)
41 \( ( 1 - p T )( 1 + p T ) \)
43 \( 1 - 61 T + p^{2} T^{2} \)
47 \( ( 1 - p T )( 1 + p T ) \)
53 \( ( 1 - p T )( 1 + p T ) \)
59 \( ( 1 - p T )( 1 + p T ) \)
61 \( 1 - 47 T + p^{2} T^{2} \)
67 \( 1 - 109 T + p^{2} T^{2} \)
71 \( ( 1 - p T )( 1 + p T ) \)
73 \( 1 - 46 T + p^{2} T^{2} \)
79 \( 1 + 142 T + p^{2} T^{2} \)
83 \( ( 1 - p T )( 1 + p T ) \)
89 \( ( 1 - p T )( 1 + p T ) \)
97 \( 1 - 169 T + p^{2} T^{2} \)
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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

−14.42707443017962127893503757179, −13.04667991370884111735625737301, −12.46213701794879877616905149124, −10.76432070397562879502361196424, −9.808637396540701377706701613764, −8.590648228047442639180407067464, −7.15891220772788699271069501067, −6.25460592902984727996751972766, −3.72576451583680290568787903805, −2.41569492732756435213236695237, 2.41569492732756435213236695237, 3.72576451583680290568787903805, 6.25460592902984727996751972766, 7.15891220772788699271069501067, 8.590648228047442639180407067464, 9.808637396540701377706701613764, 10.76432070397562879502361196424, 12.46213701794879877616905149124, 13.04667991370884111735625737301, 14.42707443017962127893503757179

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