Properties

Label 2-1045-1.1-c5-0-152
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  + 9.31·2-s + 1.66·3-s + 54.7·4-s + 25·5-s + 15.5·6-s − 125.·7-s + 212.·8-s − 240.·9-s + 232.·10-s − 121·11-s + 91.2·12-s + 837.·13-s − 1.16e3·14-s + 41.6·15-s + 222.·16-s + 760.·17-s − 2.23e3·18-s − 361·19-s + 1.36e3·20-s − 209.·21-s − 1.12e3·22-s + 2.71e3·23-s + 353.·24-s + 625·25-s + 7.80e3·26-s − 805.·27-s − 6.87e3·28-s + ⋯
L(s)  = 1  + 1.64·2-s + 0.106·3-s + 1.71·4-s + 0.447·5-s + 0.175·6-s − 0.968·7-s + 1.17·8-s − 0.988·9-s + 0.736·10-s − 0.301·11-s + 0.182·12-s + 1.37·13-s − 1.59·14-s + 0.0477·15-s + 0.217·16-s + 0.637·17-s − 1.62·18-s − 0.229·19-s + 0.765·20-s − 0.103·21-s − 0.496·22-s + 1.07·23-s + 0.125·24-s + 0.200·25-s + 2.26·26-s − 0.212·27-s − 1.65·28-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

\(L(3)\) \(\approx\) \(6.466748617\)
\(L(\frac12)\) \(\approx\) \(6.466748617\)
\(L(\frac{7}{2})\) 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 \( 1 - 9.31T + 32T^{2} \)
3 \( 1 - 1.66T + 243T^{2} \)
7 \( 1 + 125.T + 1.68e4T^{2} \)
13 \( 1 - 837.T + 3.71e5T^{2} \)
17 \( 1 - 760.T + 1.41e6T^{2} \)
23 \( 1 - 2.71e3T + 6.43e6T^{2} \)
29 \( 1 - 4.39e3T + 2.05e7T^{2} \)
31 \( 1 - 6.35e3T + 2.86e7T^{2} \)
37 \( 1 + 1.21e4T + 6.93e7T^{2} \)
41 \( 1 + 5.60e3T + 1.15e8T^{2} \)
43 \( 1 - 1.79e4T + 1.47e8T^{2} \)
47 \( 1 - 1.59e4T + 2.29e8T^{2} \)
53 \( 1 - 2.74e4T + 4.18e8T^{2} \)
59 \( 1 - 3.74e4T + 7.14e8T^{2} \)
61 \( 1 + 1.76e4T + 8.44e8T^{2} \)
67 \( 1 - 7.24e3T + 1.35e9T^{2} \)
71 \( 1 + 5.23e4T + 1.80e9T^{2} \)
73 \( 1 - 7.25e4T + 2.07e9T^{2} \)
79 \( 1 - 7.33e4T + 3.07e9T^{2} \)
83 \( 1 + 5.73e4T + 3.93e9T^{2} \)
89 \( 1 - 1.36e5T + 5.58e9T^{2} \)
97 \( 1 + 1.43e5T + 8.58e9T^{2} \)
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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

−9.093810196091767434749669873923, −8.418416667540054785475328231794, −7.04520711024655733654913838256, −6.28019994813547041314658323102, −5.76223743407900888070795856742, −4.96728986465308889205848599391, −3.75651937664434422840856572170, −3.12611901638546003070961602606, −2.39245059675302670393470141901, −0.841027160904928353758228923224, 0.841027160904928353758228923224, 2.39245059675302670393470141901, 3.12611901638546003070961602606, 3.75651937664434422840856572170, 4.96728986465308889205848599391, 5.76223743407900888070795856742, 6.28019994813547041314658323102, 7.04520711024655733654913838256, 8.418416667540054785475328231794, 9.093810196091767434749669873923

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