Properties

Label 2-75-3.2-c10-0-47
Degree $2$
Conductor $75$
Sign $1$
Analytic cond. $47.6517$
Root an. cond. $6.90302$
Motivic weight $10$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 243·3-s + 1.02e3·4-s + 3.29e4·7-s + 5.90e4·9-s + 2.48e5·12-s + 1.41e5·13-s + 1.04e6·16-s − 2.02e6·19-s + 8.01e6·21-s + 1.43e7·27-s + 3.37e7·28-s − 4.98e7·31-s + 6.04e7·36-s − 1.35e8·37-s + 3.44e7·39-s − 2.11e8·43-s + 2.54e8·48-s + 8.05e8·49-s + 1.45e8·52-s − 4.91e8·57-s + 1.55e9·61-s + 1.94e9·63-s + 1.07e9·64-s − 2.69e9·67-s + 4.14e9·73-s − 2.07e9·76-s + 3.95e9·79-s + ⋯
L(s)  = 1  + 3-s + 4-s + 1.96·7-s + 9-s + 12-s + 0.382·13-s + 16-s − 0.817·19-s + 1.96·21-s + 27-s + 1.96·28-s − 1.74·31-s + 36-s − 1.94·37-s + 0.382·39-s − 1.43·43-s + 48-s + 2.85·49-s + 0.382·52-s − 0.817·57-s + 1.83·61-s + 1.96·63-s + 64-s − 1.99·67-s + 1.99·73-s − 0.817·76-s + 1.28·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(47.6517\)
Root analytic conductor: \(6.90302\)
Motivic weight: \(10\)
Rational: yes
Arithmetic: yes
Character: $\chi_{75} (26, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :5),\ 1)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(5.028248668\)
\(L(\frac12)\) \(\approx\) \(5.028248668\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - p^{5} T \)
5 \( 1 \)
good2 \( ( 1 - p^{5} T )( 1 + p^{5} T ) \)
7 \( 1 - 32989 T + p^{10} T^{2} \)
11 \( ( 1 - p^{5} T )( 1 + p^{5} T ) \)
13 \( 1 - 141961 T + p^{10} T^{2} \)
17 \( ( 1 - p^{5} T )( 1 + p^{5} T ) \)
19 \( 1 + 2024677 T + p^{10} T^{2} \)
23 \( ( 1 - p^{5} T )( 1 + p^{5} T ) \)
29 \( ( 1 - p^{5} T )( 1 + p^{5} T ) \)
31 \( 1 + 49843573 T + p^{10} T^{2} \)
37 \( 1 + 135214586 T + p^{10} T^{2} \)
41 \( ( 1 - p^{5} T )( 1 + p^{5} T ) \)
43 \( 1 + 211108739 T + p^{10} T^{2} \)
47 \( ( 1 - p^{5} T )( 1 + p^{5} T ) \)
53 \( ( 1 - p^{5} T )( 1 + p^{5} T ) \)
59 \( ( 1 - p^{5} T )( 1 + p^{5} T ) \)
61 \( 1 - 1551490727 T + p^{10} T^{2} \)
67 \( 1 + 2698325411 T + p^{10} T^{2} \)
71 \( ( 1 - p^{5} T )( 1 + p^{5} T ) \)
73 \( 1 - 4144040686 T + p^{10} T^{2} \)
79 \( 1 - 3959005298 T + p^{10} T^{2} \)
83 \( ( 1 - p^{5} T )( 1 + p^{5} T ) \)
89 \( ( 1 - p^{5} T )( 1 + p^{5} T ) \)
97 \( 1 + 14411495111 T + p^{10} T^{2} \)
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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

−12.37833144881692155986123351825, −11.24420675523245592461030330274, −10.45055362563126616712256836535, −8.730990350186879091971335828613, −7.949896412814435547283956309339, −6.95018064005827930731122951372, −5.18936021317424694799352205295, −3.72629272830539077750574981420, −2.13300652955838032156945505409, −1.49326346184771924766900504360, 1.49326346184771924766900504360, 2.13300652955838032156945505409, 3.72629272830539077750574981420, 5.18936021317424694799352205295, 6.95018064005827930731122951372, 7.949896412814435547283956309339, 8.730990350186879091971335828613, 10.45055362563126616712256836535, 11.24420675523245592461030330274, 12.37833144881692155986123351825

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