Properties

Label 2-1075-1.1-c5-0-138
Degree $2$
Conductor $1075$
Sign $1$
Analytic cond. $172.412$
Root an. cond. $13.1305$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7.06·2-s + 9.19·3-s + 17.8·4-s + 64.9·6-s + 4.24·7-s − 99.7·8-s − 158.·9-s − 99.8·11-s + 164.·12-s + 441.·13-s + 29.9·14-s − 1.27e3·16-s + 1.03e3·17-s − 1.11e3·18-s + 402.·19-s + 39.0·21-s − 705.·22-s + 3.45e3·23-s − 916.·24-s + 3.11e3·26-s − 3.69e3·27-s + 75.9·28-s − 4.26e3·29-s + 3.77e3·31-s − 5.82e3·32-s − 918.·33-s + 7.29e3·34-s + ⋯
L(s)  = 1  + 1.24·2-s + 0.589·3-s + 0.558·4-s + 0.736·6-s + 0.0327·7-s − 0.551·8-s − 0.652·9-s − 0.248·11-s + 0.329·12-s + 0.724·13-s + 0.0409·14-s − 1.24·16-s + 0.866·17-s − 0.814·18-s + 0.255·19-s + 0.0193·21-s − 0.310·22-s + 1.36·23-s − 0.324·24-s + 0.903·26-s − 0.974·27-s + 0.0183·28-s − 0.942·29-s + 0.705·31-s − 1.00·32-s − 0.146·33-s + 1.08·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1075\)    =    \(5^{2} \cdot 43\)
Sign: $1$
Analytic conductor: \(172.412\)
Root analytic conductor: \(13.1305\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1075,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(4.973079604\)
\(L(\frac12)\) \(\approx\) \(4.973079604\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
43 \( 1 - 1.84e3T \)
good2 \( 1 - 7.06T + 32T^{2} \)
3 \( 1 - 9.19T + 243T^{2} \)
7 \( 1 - 4.24T + 1.68e4T^{2} \)
11 \( 1 + 99.8T + 1.61e5T^{2} \)
13 \( 1 - 441.T + 3.71e5T^{2} \)
17 \( 1 - 1.03e3T + 1.41e6T^{2} \)
19 \( 1 - 402.T + 2.47e6T^{2} \)
23 \( 1 - 3.45e3T + 6.43e6T^{2} \)
29 \( 1 + 4.26e3T + 2.05e7T^{2} \)
31 \( 1 - 3.77e3T + 2.86e7T^{2} \)
37 \( 1 - 7.57e3T + 6.93e7T^{2} \)
41 \( 1 + 1.41e4T + 1.15e8T^{2} \)
47 \( 1 - 1.34e4T + 2.29e8T^{2} \)
53 \( 1 - 1.17e4T + 4.18e8T^{2} \)
59 \( 1 - 2.39e4T + 7.14e8T^{2} \)
61 \( 1 - 4.46e4T + 8.44e8T^{2} \)
67 \( 1 + 3.92e4T + 1.35e9T^{2} \)
71 \( 1 - 6.59e3T + 1.80e9T^{2} \)
73 \( 1 - 3.63e4T + 2.07e9T^{2} \)
79 \( 1 - 6.18e4T + 3.07e9T^{2} \)
83 \( 1 - 6.46e4T + 3.93e9T^{2} \)
89 \( 1 - 2.95e4T + 5.58e9T^{2} \)
97 \( 1 + 1.93e4T + 8.58e9T^{2} \)
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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

−9.025663481802541899646142539938, −8.391013532553345342453992239401, −7.42670332962115735299998951688, −6.35431494861942862227628470221, −5.56636892053453137913065959387, −4.89214454849644550949578307867, −3.69099288019844061827511080811, −3.20066780697000665616127588759, −2.25632452774771841906280469033, −0.75888273785226763557066227916, 0.75888273785226763557066227916, 2.25632452774771841906280469033, 3.20066780697000665616127588759, 3.69099288019844061827511080811, 4.89214454849644550949578307867, 5.56636892053453137913065959387, 6.35431494861942862227628470221, 7.42670332962115735299998951688, 8.391013532553345342453992239401, 9.025663481802541899646142539938

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