Properties

Label 2-75-1.1-c3-0-2
Degree $2$
Conductor $75$
Sign $1$
Analytic cond. $4.42514$
Root an. cond. $2.10360$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.35·2-s + 3·3-s + 20.7·4-s − 16.0·6-s − 4.43·7-s − 68.1·8-s + 9·9-s − 3.43·11-s + 62.1·12-s + 78.7·13-s + 23.7·14-s + 199.·16-s + 53.1·17-s − 48.2·18-s + 20.4·19-s − 13.3·21-s + 18.4·22-s + 118.·23-s − 204.·24-s − 421.·26-s + 27·27-s − 91.8·28-s + 168.·29-s − 61.0·31-s − 523.·32-s − 10.3·33-s − 284.·34-s + ⋯
L(s)  = 1  − 1.89·2-s + 0.577·3-s + 2.58·4-s − 1.09·6-s − 0.239·7-s − 3.01·8-s + 0.333·9-s − 0.0941·11-s + 1.49·12-s + 1.67·13-s + 0.453·14-s + 3.11·16-s + 0.758·17-s − 0.631·18-s + 0.246·19-s − 0.138·21-s + 0.178·22-s + 1.07·23-s − 1.73·24-s − 3.18·26-s + 0.192·27-s − 0.620·28-s + 1.07·29-s − 0.353·31-s − 2.89·32-s − 0.0543·33-s − 1.43·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(4.42514\)
Root analytic conductor: \(2.10360\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8193841598\)
\(L(\frac12)\) \(\approx\) \(0.8193841598\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3T \)
5 \( 1 \)
good2 \( 1 + 5.35T + 8T^{2} \)
7 \( 1 + 4.43T + 343T^{2} \)
11 \( 1 + 3.43T + 1.33e3T^{2} \)
13 \( 1 - 78.7T + 2.19e3T^{2} \)
17 \( 1 - 53.1T + 4.91e3T^{2} \)
19 \( 1 - 20.4T + 6.85e3T^{2} \)
23 \( 1 - 118.T + 1.21e4T^{2} \)
29 \( 1 - 168.T + 2.43e4T^{2} \)
31 \( 1 + 61.0T + 2.97e4T^{2} \)
37 \( 1 + 246.T + 5.06e4T^{2} \)
41 \( 1 - 422.T + 6.89e4T^{2} \)
43 \( 1 - 362.T + 7.95e4T^{2} \)
47 \( 1 + 170.T + 1.03e5T^{2} \)
53 \( 1 + 546.T + 1.48e5T^{2} \)
59 \( 1 + 216.T + 2.05e5T^{2} \)
61 \( 1 - 130.T + 2.26e5T^{2} \)
67 \( 1 + 614.T + 3.00e5T^{2} \)
71 \( 1 - 324.T + 3.57e5T^{2} \)
73 \( 1 + 88.8T + 3.89e5T^{2} \)
79 \( 1 + 1.13e3T + 4.93e5T^{2} \)
83 \( 1 + 758.T + 5.71e5T^{2} \)
89 \( 1 - 195.T + 7.04e5T^{2} \)
97 \( 1 - 521T + 9.12e5T^{2} \)
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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

−14.29167349720641138117828163268, −12.74100189325988967362544988474, −11.31282448775903488562453309718, −10.42879373237086755428567101908, −9.297601093707357729820412345387, −8.509584476741105792359814000800, −7.47469207047706828605745659250, −6.20796267429547389008604065886, −3.11584581666069928201061840544, −1.23234670314887101005053616116, 1.23234670314887101005053616116, 3.11584581666069928201061840544, 6.20796267429547389008604065886, 7.47469207047706828605745659250, 8.509584476741105792359814000800, 9.297601093707357729820412345387, 10.42879373237086755428567101908, 11.31282448775903488562453309718, 12.74100189325988967362544988474, 14.29167349720641138117828163268

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