Properties

Label 2-1045-1.1-c5-0-100
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  − 10.7·2-s − 11.3·3-s + 83.8·4-s + 25·5-s + 122.·6-s − 59.3·7-s − 557.·8-s − 114.·9-s − 269.·10-s + 121·11-s − 951.·12-s + 781.·13-s + 638.·14-s − 283.·15-s + 3.32e3·16-s + 774.·17-s + 1.22e3·18-s + 361·19-s + 2.09e3·20-s + 673.·21-s − 1.30e3·22-s + 2.90e3·23-s + 6.33e3·24-s + 625·25-s − 8.40e3·26-s + 4.05e3·27-s − 4.97e3·28-s + ⋯
L(s)  = 1  − 1.90·2-s − 0.728·3-s + 2.61·4-s + 0.447·5-s + 1.38·6-s − 0.457·7-s − 3.08·8-s − 0.469·9-s − 0.850·10-s + 0.301·11-s − 1.90·12-s + 1.28·13-s + 0.870·14-s − 0.325·15-s + 3.24·16-s + 0.649·17-s + 0.893·18-s + 0.229·19-s + 1.17·20-s + 0.333·21-s − 0.573·22-s + 1.14·23-s + 2.24·24-s + 0.200·25-s − 2.43·26-s + 1.07·27-s − 1.19·28-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: $\chi_{1045} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1045,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(0.8232984147\)
\(L(\frac12)\) \(\approx\) \(0.8232984147\)
\(L(\frac{7}{2})\) 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 \( 1 + 10.7T + 32T^{2} \)
3 \( 1 + 11.3T + 243T^{2} \)
7 \( 1 + 59.3T + 1.68e4T^{2} \)
13 \( 1 - 781.T + 3.71e5T^{2} \)
17 \( 1 - 774.T + 1.41e6T^{2} \)
23 \( 1 - 2.90e3T + 6.43e6T^{2} \)
29 \( 1 + 2.59e3T + 2.05e7T^{2} \)
31 \( 1 - 2.38e3T + 2.86e7T^{2} \)
37 \( 1 - 1.23e3T + 6.93e7T^{2} \)
41 \( 1 - 8.93e3T + 1.15e8T^{2} \)
43 \( 1 + 2.98e3T + 1.47e8T^{2} \)
47 \( 1 - 2.31e4T + 2.29e8T^{2} \)
53 \( 1 - 1.76e4T + 4.18e8T^{2} \)
59 \( 1 + 1.57e4T + 7.14e8T^{2} \)
61 \( 1 - 4.66e4T + 8.44e8T^{2} \)
67 \( 1 + 2.49e4T + 1.35e9T^{2} \)
71 \( 1 - 6.61e3T + 1.80e9T^{2} \)
73 \( 1 - 6.66e4T + 2.07e9T^{2} \)
79 \( 1 - 5.48e4T + 3.07e9T^{2} \)
83 \( 1 - 5.25e4T + 3.93e9T^{2} \)
89 \( 1 + 2.77e4T + 5.58e9T^{2} \)
97 \( 1 - 1.95e4T + 8.58e9T^{2} \)
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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

−9.148921739100589671797925420576, −8.620320946856510241019492385534, −7.67849645983905555999799911095, −6.71184579985472273299159780878, −6.15660871716597498550084230537, −5.40771739888719749194970654666, −3.48471289738462765370111272511, −2.48469276119001125621601300595, −1.20035607090913465651494966823, −0.64617400699796922027558597647, 0.64617400699796922027558597647, 1.20035607090913465651494966823, 2.48469276119001125621601300595, 3.48471289738462765370111272511, 5.40771739888719749194970654666, 6.15660871716597498550084230537, 6.71184579985472273299159780878, 7.67849645983905555999799911095, 8.620320946856510241019492385534, 9.148921739100589671797925420576

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