Properties

Label 2-1575-1.1-c3-0-117
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  + 5.41·2-s + 21.3·4-s + 7·7-s + 72.0·8-s + 52.2·11-s − 30.6·13-s + 37.8·14-s + 219.·16-s + 37.2·17-s + 80.2·19-s + 282.·22-s + 25.8·23-s − 165.·26-s + 149.·28-s − 20.9·29-s − 314.·31-s + 613.·32-s + 201.·34-s − 197.·37-s + 434.·38-s − 11.3·41-s + 33.8·43-s + 1.11e3·44-s + 139.·46-s − 361.·47-s + 49·49-s − 653.·52-s + ⋯
L(s)  = 1  + 1.91·2-s + 2.66·4-s + 0.377·7-s + 3.18·8-s + 1.43·11-s − 0.654·13-s + 0.723·14-s + 3.43·16-s + 0.531·17-s + 0.968·19-s + 2.74·22-s + 0.234·23-s − 1.25·26-s + 1.00·28-s − 0.134·29-s − 1.82·31-s + 3.38·32-s + 1.01·34-s − 0.875·37-s + 1.85·38-s − 0.0432·41-s + 0.119·43-s + 3.81·44-s + 0.448·46-s − 1.12·47-s + 0.142·49-s − 1.74·52-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

\(L(2)\) \(\approx\) \(9.758534075\)
\(L(\frac12)\) \(\approx\) \(9.758534075\)
\(L(\frac{5}{2})\) 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 \( 1 - 5.41T + 8T^{2} \)
11 \( 1 - 52.2T + 1.33e3T^{2} \)
13 \( 1 + 30.6T + 2.19e3T^{2} \)
17 \( 1 - 37.2T + 4.91e3T^{2} \)
19 \( 1 - 80.2T + 6.85e3T^{2} \)
23 \( 1 - 25.8T + 1.21e4T^{2} \)
29 \( 1 + 20.9T + 2.43e4T^{2} \)
31 \( 1 + 314.T + 2.97e4T^{2} \)
37 \( 1 + 197.T + 5.06e4T^{2} \)
41 \( 1 + 11.3T + 6.89e4T^{2} \)
43 \( 1 - 33.8T + 7.95e4T^{2} \)
47 \( 1 + 361.T + 1.03e5T^{2} \)
53 \( 1 - 153.T + 1.48e5T^{2} \)
59 \( 1 - 616T + 2.05e5T^{2} \)
61 \( 1 - 15.2T + 2.26e5T^{2} \)
67 \( 1 - 166.T + 3.00e5T^{2} \)
71 \( 1 - 952T + 3.57e5T^{2} \)
73 \( 1 - 148.T + 3.89e5T^{2} \)
79 \( 1 - 857.T + 4.93e5T^{2} \)
83 \( 1 - 660.T + 5.71e5T^{2} \)
89 \( 1 - 45.7T + 7.04e5T^{2} \)
97 \( 1 + 1.68e3T + 9.12e5T^{2} \)
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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

−9.137782411391100420742336774239, −7.84870358613648712482921814105, −7.09503321598802074787734856127, −6.51244894676435169638107792322, −5.43981683249777740638572004133, −5.04026255138158721014435521234, −3.89834350273012229687821583878, −3.44528785216566537713760772089, −2.21941165757773480385632616640, −1.28656423209930191114592692801, 1.28656423209930191114592692801, 2.21941165757773480385632616640, 3.44528785216566537713760772089, 3.89834350273012229687821583878, 5.04026255138158721014435521234, 5.43981683249777740638572004133, 6.51244894676435169638107792322, 7.09503321598802074787734856127, 7.84870358613648712482921814105, 9.137782411391100420742336774239

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