Properties

Label 2-1045-1.1-c3-0-100
Degree $2$
Conductor $1045$
Sign $1$
Analytic cond. $61.6569$
Root an. cond. $7.85219$
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  + 2.47·2-s + 8.18·3-s − 1.88·4-s − 5·5-s + 20.2·6-s + 19.1·7-s − 24.4·8-s + 40.0·9-s − 12.3·10-s − 11·11-s − 15.4·12-s + 27.8·13-s + 47.4·14-s − 40.9·15-s − 45.3·16-s + 9.83·17-s + 99.0·18-s − 19·19-s + 9.43·20-s + 156.·21-s − 27.1·22-s + 153.·23-s − 200.·24-s + 25·25-s + 68.9·26-s + 106.·27-s − 36.1·28-s + ⋯
L(s)  = 1  + 0.874·2-s + 1.57·3-s − 0.235·4-s − 0.447·5-s + 1.37·6-s + 1.03·7-s − 1.08·8-s + 1.48·9-s − 0.390·10-s − 0.301·11-s − 0.371·12-s + 0.594·13-s + 0.904·14-s − 0.704·15-s − 0.708·16-s + 0.140·17-s + 1.29·18-s − 0.229·19-s + 0.105·20-s + 1.63·21-s − 0.263·22-s + 1.38·23-s − 1.70·24-s + 0.200·25-s + 0.519·26-s + 0.761·27-s − 0.244·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $1$
Analytic conductor: \(61.6569\)
Root analytic conductor: \(7.85219\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1045,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(5.542285294\)
\(L(\frac12)\) \(\approx\) \(5.542285294\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + 5T \)
11 \( 1 + 11T \)
19 \( 1 + 19T \)
good2 \( 1 - 2.47T + 8T^{2} \)
3 \( 1 - 8.18T + 27T^{2} \)
7 \( 1 - 19.1T + 343T^{2} \)
13 \( 1 - 27.8T + 2.19e3T^{2} \)
17 \( 1 - 9.83T + 4.91e3T^{2} \)
23 \( 1 - 153.T + 1.21e4T^{2} \)
29 \( 1 - 151.T + 2.43e4T^{2} \)
31 \( 1 - 196.T + 2.97e4T^{2} \)
37 \( 1 - 379.T + 5.06e4T^{2} \)
41 \( 1 + 26.9T + 6.89e4T^{2} \)
43 \( 1 - 27.7T + 7.95e4T^{2} \)
47 \( 1 - 148.T + 1.03e5T^{2} \)
53 \( 1 - 348.T + 1.48e5T^{2} \)
59 \( 1 + 110.T + 2.05e5T^{2} \)
61 \( 1 + 45.4T + 2.26e5T^{2} \)
67 \( 1 + 940.T + 3.00e5T^{2} \)
71 \( 1 + 36.8T + 3.57e5T^{2} \)
73 \( 1 + 838.T + 3.89e5T^{2} \)
79 \( 1 - 267.T + 4.93e5T^{2} \)
83 \( 1 + 417.T + 5.71e5T^{2} \)
89 \( 1 + 1.10e3T + 7.04e5T^{2} \)
97 \( 1 - 705.T + 9.12e5T^{2} \)
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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

−9.226362825186630202654860850057, −8.578187891953467117592431104415, −8.112379837067840964243374675858, −7.23004421605609693029134764412, −5.98815064705310693361504316681, −4.74927982714456701692248801239, −4.28823336092267080774570431489, −3.22932179513558334855845407803, −2.57422757473426497905326680339, −1.09349691265884816485174284525, 1.09349691265884816485174284525, 2.57422757473426497905326680339, 3.22932179513558334855845407803, 4.28823336092267080774570431489, 4.74927982714456701692248801239, 5.98815064705310693361504316681, 7.23004421605609693029134764412, 8.112379837067840964243374675858, 8.578187891953467117592431104415, 9.226362825186630202654860850057

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