Properties

Label 2-1575-1.1-c1-0-1
Degree $2$
Conductor $1575$
Sign $1$
Analytic cond. $12.5764$
Root an. cond. $3.54632$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.34·2-s − 0.192·4-s − 7-s + 2.94·8-s − 5.63·11-s − 4.38·13-s + 1.34·14-s − 3.57·16-s − 2.68·17-s + 8.38·19-s + 7.57·22-s − 5.63·23-s + 5.89·26-s + 0.192·28-s + 8.32·29-s + 6·31-s − 1.08·32-s + 3.61·34-s + 3·37-s − 11.2·38-s − 5.37·41-s + 1.38·43-s + 1.08·44-s + 7.57·46-s − 8.58·47-s + 49-s + 0.844·52-s + ⋯
L(s)  = 1  − 0.950·2-s − 0.0962·4-s − 0.377·7-s + 1.04·8-s − 1.69·11-s − 1.21·13-s + 0.359·14-s − 0.894·16-s − 0.652·17-s + 1.92·19-s + 1.61·22-s − 1.17·23-s + 1.15·26-s + 0.0363·28-s + 1.54·29-s + 1.07·31-s − 0.191·32-s + 0.619·34-s + 0.493·37-s − 1.82·38-s − 0.839·41-s + 0.211·43-s + 0.163·44-s + 1.11·46-s − 1.25·47-s + 0.142·49-s + 0.117·52-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(12.5764\)
Root analytic conductor: \(3.54632\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1575,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5555126606\)
\(L(\frac12)\) \(\approx\) \(0.5555126606\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
good2 \( 1 + 1.34T + 2T^{2} \)
11 \( 1 + 5.63T + 11T^{2} \)
13 \( 1 + 4.38T + 13T^{2} \)
17 \( 1 + 2.68T + 17T^{2} \)
19 \( 1 - 8.38T + 19T^{2} \)
23 \( 1 + 5.63T + 23T^{2} \)
29 \( 1 - 8.32T + 29T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 - 3T + 37T^{2} \)
41 \( 1 + 5.37T + 41T^{2} \)
43 \( 1 - 1.38T + 43T^{2} \)
47 \( 1 + 8.58T + 47T^{2} \)
53 \( 1 - 5.37T + 53T^{2} \)
59 \( 1 + 8.58T + 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 - 1.38T + 67T^{2} \)
71 \( 1 - 0.258T + 71T^{2} \)
73 \( 1 + 6.38T + 73T^{2} \)
79 \( 1 - 5.38T + 79T^{2} \)
83 \( 1 - 16.6T + 83T^{2} \)
89 \( 1 - 13.9T + 89T^{2} \)
97 \( 1 - 10.7T + 97T^{2} \)
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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

−9.598531203176842568861868815095, −8.613868083121964384454364587508, −7.80584826556410665932381719813, −7.47172745989270748281450749425, −6.34027337238007324559786609945, −5.10049941665954276071404646082, −4.68356009362406799924262241932, −3.15898221952644443036656883654, −2.21246428365054150094463502862, −0.58376036990521719785602097548, 0.58376036990521719785602097548, 2.21246428365054150094463502862, 3.15898221952644443036656883654, 4.68356009362406799924262241932, 5.10049941665954276071404646082, 6.34027337238007324559786609945, 7.47172745989270748281450749425, 7.80584826556410665932381719813, 8.613868083121964384454364587508, 9.598531203176842568861868815095

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