Properties

Label 2-1045-1.1-c1-0-8
Degree $2$
Conductor $1045$
Sign $1$
Analytic cond. $8.34436$
Root an. cond. $2.88866$
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  − 1.08·2-s − 0.870·3-s − 0.829·4-s − 5-s + 0.942·6-s + 4.02·7-s + 3.06·8-s − 2.24·9-s + 1.08·10-s + 11-s + 0.722·12-s − 3.57·13-s − 4.35·14-s + 0.870·15-s − 1.65·16-s + 2.13·17-s + 2.42·18-s − 19-s + 0.829·20-s − 3.50·21-s − 1.08·22-s − 1.52·23-s − 2.66·24-s + 25-s + 3.86·26-s + 4.56·27-s − 3.34·28-s + ⋯
L(s)  = 1  − 0.764·2-s − 0.502·3-s − 0.414·4-s − 0.447·5-s + 0.384·6-s + 1.52·7-s + 1.08·8-s − 0.747·9-s + 0.342·10-s + 0.301·11-s + 0.208·12-s − 0.990·13-s − 1.16·14-s + 0.224·15-s − 0.413·16-s + 0.518·17-s + 0.571·18-s − 0.229·19-s + 0.185·20-s − 0.765·21-s − 0.230·22-s − 0.318·23-s − 0.544·24-s + 0.200·25-s + 0.757·26-s + 0.878·27-s − 0.631·28-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $1$
Analytic conductor: \(8.34436\)
Root analytic conductor: \(2.88866\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1045,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6997063489\)
\(L(\frac12)\) \(\approx\) \(0.6997063489\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + T \)
11 \( 1 - T \)
19 \( 1 + T \)
good2 \( 1 + 1.08T + 2T^{2} \)
3 \( 1 + 0.870T + 3T^{2} \)
7 \( 1 - 4.02T + 7T^{2} \)
13 \( 1 + 3.57T + 13T^{2} \)
17 \( 1 - 2.13T + 17T^{2} \)
23 \( 1 + 1.52T + 23T^{2} \)
29 \( 1 - 0.640T + 29T^{2} \)
31 \( 1 + 2.79T + 31T^{2} \)
37 \( 1 - 6.88T + 37T^{2} \)
41 \( 1 - 2.11T + 41T^{2} \)
43 \( 1 + 11.2T + 43T^{2} \)
47 \( 1 + 5.49T + 47T^{2} \)
53 \( 1 - 10.3T + 53T^{2} \)
59 \( 1 - 10.5T + 59T^{2} \)
61 \( 1 - 7.73T + 61T^{2} \)
67 \( 1 - 7.97T + 67T^{2} \)
71 \( 1 - 7.60T + 71T^{2} \)
73 \( 1 - 3.48T + 73T^{2} \)
79 \( 1 - 5.50T + 79T^{2} \)
83 \( 1 - 15.4T + 83T^{2} \)
89 \( 1 - 17.3T + 89T^{2} \)
97 \( 1 + 6.28T + 97T^{2} \)
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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

−9.923231944485780332127927229346, −8.987760487258380191091899924122, −8.155537935668601371253205151891, −7.83700667095599379858049702343, −6.74921370186553511514199366657, −5.32461967376201100596651681454, −4.90604535936812251423652538293, −3.85205684256680292528431862841, −2.14389531229314839874012979233, −0.75391757395635986766845305494, 0.75391757395635986766845305494, 2.14389531229314839874012979233, 3.85205684256680292528431862841, 4.90604535936812251423652538293, 5.32461967376201100596651681454, 6.74921370186553511514199366657, 7.83700667095599379858049702343, 8.155537935668601371253205151891, 8.987760487258380191091899924122, 9.923231944485780332127927229346

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