Properties

Label 2-1045-1.1-c5-0-3
Degree $2$
Conductor $1045$
Sign $1$
Analytic cond. $167.601$
Root an. cond. $12.9460$
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  + 7.46·2-s − 24.9·3-s + 23.7·4-s − 25·5-s − 186.·6-s − 45.5·7-s − 61.5·8-s + 377.·9-s − 186.·10-s + 121·11-s − 591.·12-s − 988.·13-s − 339.·14-s + 622.·15-s − 1.21e3·16-s − 633.·17-s + 2.81e3·18-s − 361·19-s − 593.·20-s + 1.13e3·21-s + 903.·22-s − 3.18e3·23-s + 1.53e3·24-s + 625·25-s − 7.37e3·26-s − 3.35e3·27-s − 1.08e3·28-s + ⋯
L(s)  = 1  + 1.32·2-s − 1.59·3-s + 0.742·4-s − 0.447·5-s − 2.10·6-s − 0.351·7-s − 0.339·8-s + 1.55·9-s − 0.590·10-s + 0.301·11-s − 1.18·12-s − 1.62·13-s − 0.463·14-s + 0.714·15-s − 1.19·16-s − 0.531·17-s + 2.05·18-s − 0.229·19-s − 0.332·20-s + 0.561·21-s + 0.398·22-s − 1.25·23-s + 0.543·24-s + 0.200·25-s − 2.14·26-s − 0.884·27-s − 0.260·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $1$
Analytic conductor: \(167.601\)
Root analytic conductor: \(12.9460\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1045,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(0.01928124904\)
\(L(\frac12)\) \(\approx\) \(0.01928124904\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + 25T \)
11 \( 1 - 121T \)
19 \( 1 + 361T \)
good2 \( 1 - 7.46T + 32T^{2} \)
3 \( 1 + 24.9T + 243T^{2} \)
7 \( 1 + 45.5T + 1.68e4T^{2} \)
13 \( 1 + 988.T + 3.71e5T^{2} \)
17 \( 1 + 633.T + 1.41e6T^{2} \)
23 \( 1 + 3.18e3T + 6.43e6T^{2} \)
29 \( 1 + 1.94e3T + 2.05e7T^{2} \)
31 \( 1 + 223.T + 2.86e7T^{2} \)
37 \( 1 + 4.76e3T + 6.93e7T^{2} \)
41 \( 1 + 3.15e3T + 1.15e8T^{2} \)
43 \( 1 + 2.39e4T + 1.47e8T^{2} \)
47 \( 1 + 2.56e4T + 2.29e8T^{2} \)
53 \( 1 + 1.37e4T + 4.18e8T^{2} \)
59 \( 1 - 4.81e3T + 7.14e8T^{2} \)
61 \( 1 - 2.53e4T + 8.44e8T^{2} \)
67 \( 1 + 3.28e3T + 1.35e9T^{2} \)
71 \( 1 + 3.38e4T + 1.80e9T^{2} \)
73 \( 1 - 2.74e4T + 2.07e9T^{2} \)
79 \( 1 - 3.42e4T + 3.07e9T^{2} \)
83 \( 1 + 2.40e4T + 3.93e9T^{2} \)
89 \( 1 - 7.67e4T + 5.58e9T^{2} \)
97 \( 1 + 1.16e5T + 8.58e9T^{2} \)
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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

−9.514015328925518202630115765467, −8.134217979628084158944264282941, −6.84903015094715002283271213788, −6.56172354326666856491436094114, −5.53710038655686607276373450992, −4.91075107812877068323567600874, −4.28532790096341322637879448481, −3.25994604836089932457172044478, −1.91543587799810754147567336947, −0.05012531932897915162140909335, 0.05012531932897915162140909335, 1.91543587799810754147567336947, 3.25994604836089932457172044478, 4.28532790096341322637879448481, 4.91075107812877068323567600874, 5.53710038655686607276373450992, 6.56172354326666856491436094114, 6.84903015094715002283271213788, 8.134217979628084158944264282941, 9.514015328925518202630115765467

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