Properties

Label 2-1575-1.1-c3-0-59
Degree $2$
Conductor $1575$
Sign $1$
Analytic cond. $92.9280$
Root an. cond. $9.63991$
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  − 3.53·2-s + 4.46·4-s + 7·7-s + 12.4·8-s + 2.93·11-s + 19.0·13-s − 24.7·14-s − 79.7·16-s + 122.·17-s + 107.·19-s − 10.3·22-s + 210.·23-s − 67.3·26-s + 31.2·28-s − 95.4·29-s − 94.3·31-s + 181.·32-s − 432.·34-s − 97.1·37-s − 379.·38-s + 491.·41-s + 43.0·43-s + 13.1·44-s − 743.·46-s + 473.·47-s + 49·49-s + 85.1·52-s + ⋯
L(s)  = 1  − 1.24·2-s + 0.558·4-s + 0.377·7-s + 0.551·8-s + 0.0805·11-s + 0.406·13-s − 0.471·14-s − 1.24·16-s + 1.74·17-s + 1.29·19-s − 0.100·22-s + 1.90·23-s − 0.507·26-s + 0.211·28-s − 0.611·29-s − 0.546·31-s + 1.00·32-s − 2.18·34-s − 0.431·37-s − 1.61·38-s + 1.87·41-s + 0.152·43-s + 0.0449·44-s − 2.38·46-s + 1.46·47-s + 0.142·49-s + 0.227·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(92.9280\)
Root analytic conductor: \(9.63991\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1575} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1575,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.437857769\)
\(L(\frac12)\) \(\approx\) \(1.437857769\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 - 7T \)
good2 \( 1 + 3.53T + 8T^{2} \)
11 \( 1 - 2.93T + 1.33e3T^{2} \)
13 \( 1 - 19.0T + 2.19e3T^{2} \)
17 \( 1 - 122.T + 4.91e3T^{2} \)
19 \( 1 - 107.T + 6.85e3T^{2} \)
23 \( 1 - 210.T + 1.21e4T^{2} \)
29 \( 1 + 95.4T + 2.43e4T^{2} \)
31 \( 1 + 94.3T + 2.97e4T^{2} \)
37 \( 1 + 97.1T + 5.06e4T^{2} \)
41 \( 1 - 491.T + 6.89e4T^{2} \)
43 \( 1 - 43.0T + 7.95e4T^{2} \)
47 \( 1 - 473.T + 1.03e5T^{2} \)
53 \( 1 + 183.T + 1.48e5T^{2} \)
59 \( 1 - 760.T + 2.05e5T^{2} \)
61 \( 1 + 198.T + 2.26e5T^{2} \)
67 \( 1 - 309.T + 3.00e5T^{2} \)
71 \( 1 + 665.T + 3.57e5T^{2} \)
73 \( 1 + 621.T + 3.89e5T^{2} \)
79 \( 1 + 24.7T + 4.93e5T^{2} \)
83 \( 1 + 406.T + 5.71e5T^{2} \)
89 \( 1 + 261.T + 7.04e5T^{2} \)
97 \( 1 - 1.00e3T + 9.12e5T^{2} \)
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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

−9.153353705586358740942453497971, −8.368641941146238382480236027923, −7.43914919027476010469579570411, −7.24818549978646837826075463789, −5.76931935054066266328814399972, −5.08914932039572729156213172901, −3.88725987228588153584368358120, −2.81147344313458567046704702605, −1.37364563684643671998184723910, −0.825756224556272554430617877710, 0.825756224556272554430617877710, 1.37364563684643671998184723910, 2.81147344313458567046704702605, 3.88725987228588153584368358120, 5.08914932039572729156213172901, 5.76931935054066266328814399972, 7.24818549978646837826075463789, 7.43914919027476010469579570411, 8.368641941146238382480236027923, 9.153353705586358740942453497971

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