Properties

Label 2-1045-1.1-c5-0-70
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  − 8.55·2-s + 12.0·3-s + 41.1·4-s + 25·5-s − 103.·6-s − 150.·7-s − 77.9·8-s − 97.2·9-s − 213.·10-s + 121·11-s + 496.·12-s − 71.2·13-s + 1.28e3·14-s + 301.·15-s − 649.·16-s + 933.·17-s + 831.·18-s + 361·19-s + 1.02e3·20-s − 1.81e3·21-s − 1.03e3·22-s + 2.30e3·23-s − 941.·24-s + 625·25-s + 608.·26-s − 4.10e3·27-s − 6.19e3·28-s + ⋯
L(s)  = 1  − 1.51·2-s + 0.774·3-s + 1.28·4-s + 0.447·5-s − 1.17·6-s − 1.16·7-s − 0.430·8-s − 0.400·9-s − 0.676·10-s + 0.301·11-s + 0.995·12-s − 0.116·13-s + 1.75·14-s + 0.346·15-s − 0.633·16-s + 0.783·17-s + 0.605·18-s + 0.229·19-s + 0.574·20-s − 0.899·21-s − 0.455·22-s + 0.909·23-s − 0.333·24-s + 0.200·25-s + 0.176·26-s − 1.08·27-s − 1.49·28-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

\(L(3)\) \(\approx\) \(1.052876009\)
\(L(\frac12)\) \(\approx\) \(1.052876009\)
\(L(\frac{7}{2})\) 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 \( 1 + 8.55T + 32T^{2} \)
3 \( 1 - 12.0T + 243T^{2} \)
7 \( 1 + 150.T + 1.68e4T^{2} \)
13 \( 1 + 71.2T + 3.71e5T^{2} \)
17 \( 1 - 933.T + 1.41e6T^{2} \)
23 \( 1 - 2.30e3T + 6.43e6T^{2} \)
29 \( 1 - 4.09e3T + 2.05e7T^{2} \)
31 \( 1 - 5.44e3T + 2.86e7T^{2} \)
37 \( 1 - 6.72e3T + 6.93e7T^{2} \)
41 \( 1 + 1.14e4T + 1.15e8T^{2} \)
43 \( 1 + 1.36e4T + 1.47e8T^{2} \)
47 \( 1 + 2.07e4T + 2.29e8T^{2} \)
53 \( 1 + 1.92e4T + 4.18e8T^{2} \)
59 \( 1 + 4.66e4T + 7.14e8T^{2} \)
61 \( 1 - 1.68e4T + 8.44e8T^{2} \)
67 \( 1 - 4.52e4T + 1.35e9T^{2} \)
71 \( 1 + 1.63e4T + 1.80e9T^{2} \)
73 \( 1 - 9.14e3T + 2.07e9T^{2} \)
79 \( 1 - 9.23e4T + 3.07e9T^{2} \)
83 \( 1 + 1.63e4T + 3.93e9T^{2} \)
89 \( 1 - 5.62e4T + 5.58e9T^{2} \)
97 \( 1 + 9.09e4T + 8.58e9T^{2} \)
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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

−9.378623565940597842699647494922, −8.430559096732298377907094651263, −7.928127728860201124347996560106, −6.81721134356847836334799856461, −6.27661813594863823183759020448, −4.93247084025109100857631322051, −3.34571527806515005972567825685, −2.75542717947194785585080805712, −1.58590993345610302627581155983, −0.54578314610514960852036768868, 0.54578314610514960852036768868, 1.58590993345610302627581155983, 2.75542717947194785585080805712, 3.34571527806515005972567825685, 4.93247084025109100857631322051, 6.27661813594863823183759020448, 6.81721134356847836334799856461, 7.928127728860201124347996560106, 8.430559096732298377907094651263, 9.378623565940597842699647494922

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