Properties

Label 2-75-1.1-c3-0-5
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  + 3.35·2-s + 3·3-s + 3.28·4-s + 10.0·6-s + 30.4·7-s − 15.8·8-s + 9·9-s + 31.4·11-s + 9.84·12-s − 60.7·13-s + 102.·14-s − 79.4·16-s − 121.·17-s + 30.2·18-s − 14.4·19-s + 91.3·21-s + 105.·22-s + 13.6·23-s − 47.5·24-s − 204.·26-s + 27·27-s + 99.8·28-s − 76.0·29-s + 183.·31-s − 140.·32-s + 94.3·33-s − 407.·34-s + ⋯
L(s)  = 1  + 1.18·2-s + 0.577·3-s + 0.410·4-s + 0.685·6-s + 1.64·7-s − 0.700·8-s + 0.333·9-s + 0.861·11-s + 0.236·12-s − 1.29·13-s + 1.95·14-s − 1.24·16-s − 1.72·17-s + 0.395·18-s − 0.174·19-s + 0.948·21-s + 1.02·22-s + 0.124·23-s − 0.404·24-s − 1.53·26-s + 0.192·27-s + 0.674·28-s − 0.486·29-s + 1.06·31-s − 0.774·32-s + 0.497·33-s − 2.05·34-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

\(L(2)\) \(\approx\) \(3.042681408\)
\(L(\frac12)\) \(\approx\) \(3.042681408\)
\(L(\frac{5}{2})\) 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 \( 1 - 3.35T + 8T^{2} \)
7 \( 1 - 30.4T + 343T^{2} \)
11 \( 1 - 31.4T + 1.33e3T^{2} \)
13 \( 1 + 60.7T + 2.19e3T^{2} \)
17 \( 1 + 121.T + 4.91e3T^{2} \)
19 \( 1 + 14.4T + 6.85e3T^{2} \)
23 \( 1 - 13.6T + 1.21e4T^{2} \)
29 \( 1 + 76.0T + 2.43e4T^{2} \)
31 \( 1 - 183.T + 2.97e4T^{2} \)
37 \( 1 + 37.3T + 5.06e4T^{2} \)
41 \( 1 + 30.6T + 6.89e4T^{2} \)
43 \( 1 - 327.T + 7.95e4T^{2} \)
47 \( 1 + 449.T + 1.03e5T^{2} \)
53 \( 1 + 301.T + 1.48e5T^{2} \)
59 \( 1 - 340.T + 2.05e5T^{2} \)
61 \( 1 - 619.T + 2.26e5T^{2} \)
67 \( 1 - 256.T + 3.00e5T^{2} \)
71 \( 1 - 499.T + 3.57e5T^{2} \)
73 \( 1 + 19.1T + 3.89e5T^{2} \)
79 \( 1 - 257.T + 4.93e5T^{2} \)
83 \( 1 - 914.T + 5.71e5T^{2} \)
89 \( 1 + 1.05e3T + 7.04e5T^{2} \)
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

−14.24332550083983904882982269161, −13.18550028868367366655362509949, −12.01807971394803128186190110926, −11.15476230262807161802078519116, −9.358852671101762233775730606498, −8.260820253373143265103536819552, −6.77615100985806070963595453584, −5.00215006540971788479692198185, −4.20267866471486261462943912769, −2.26218906252631476550710811291, 2.26218906252631476550710811291, 4.20267866471486261462943912769, 5.00215006540971788479692198185, 6.77615100985806070963595453584, 8.260820253373143265103536819552, 9.358852671101762233775730606498, 11.15476230262807161802078519116, 12.01807971394803128186190110926, 13.18550028868367366655362509949, 14.24332550083983904882982269161

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