Properties

Label 2-1045-1.1-c5-0-39
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  − 2.78·2-s − 26.7·3-s − 24.2·4-s − 25·5-s + 74.6·6-s − 117.·7-s + 156.·8-s + 474.·9-s + 69.6·10-s − 121·11-s + 649.·12-s − 1.03e3·13-s + 328.·14-s + 669.·15-s + 338.·16-s + 1.51e3·17-s − 1.32e3·18-s + 361·19-s + 605.·20-s + 3.15e3·21-s + 337.·22-s + 1.69e3·23-s − 4.19e3·24-s + 625·25-s + 2.88e3·26-s − 6.19e3·27-s + 2.85e3·28-s + ⋯
L(s)  = 1  − 0.492·2-s − 1.71·3-s − 0.757·4-s − 0.447·5-s + 0.846·6-s − 0.909·7-s + 0.865·8-s + 1.95·9-s + 0.220·10-s − 0.301·11-s + 1.30·12-s − 1.69·13-s + 0.448·14-s + 0.768·15-s + 0.330·16-s + 1.27·17-s − 0.961·18-s + 0.229·19-s + 0.338·20-s + 1.56·21-s + 0.148·22-s + 0.667·23-s − 1.48·24-s + 0.200·25-s + 0.836·26-s − 1.63·27-s + 0.689·28-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

\(L(3)\) \(\approx\) \(0.3371538951\)
\(L(\frac12)\) \(\approx\) \(0.3371538951\)
\(L(\frac{7}{2})\) 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 \( 1 + 2.78T + 32T^{2} \)
3 \( 1 + 26.7T + 243T^{2} \)
7 \( 1 + 117.T + 1.68e4T^{2} \)
13 \( 1 + 1.03e3T + 3.71e5T^{2} \)
17 \( 1 - 1.51e3T + 1.41e6T^{2} \)
23 \( 1 - 1.69e3T + 6.43e6T^{2} \)
29 \( 1 - 4.98e3T + 2.05e7T^{2} \)
31 \( 1 - 6.81e3T + 2.86e7T^{2} \)
37 \( 1 + 840.T + 6.93e7T^{2} \)
41 \( 1 + 4.17e3T + 1.15e8T^{2} \)
43 \( 1 - 6.31e3T + 1.47e8T^{2} \)
47 \( 1 - 2.47e3T + 2.29e8T^{2} \)
53 \( 1 - 1.41e4T + 4.18e8T^{2} \)
59 \( 1 + 4.14e3T + 7.14e8T^{2} \)
61 \( 1 - 4.98e4T + 8.44e8T^{2} \)
67 \( 1 + 6.13e4T + 1.35e9T^{2} \)
71 \( 1 - 6.01e4T + 1.80e9T^{2} \)
73 \( 1 + 1.42e4T + 2.07e9T^{2} \)
79 \( 1 + 3.26e4T + 3.07e9T^{2} \)
83 \( 1 + 1.93e3T + 3.93e9T^{2} \)
89 \( 1 + 1.44e5T + 5.58e9T^{2} \)
97 \( 1 + 1.98e4T + 8.58e9T^{2} \)
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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

−9.613832720862587536655758065324, −8.300242219905904869414763619504, −7.36752038700786897524586130613, −6.76602781915938949456628218229, −5.61467486788051575201457821394, −5.00461835249379915169013816868, −4.26453259039238897813000245729, −2.93749784307262355008325067685, −1.06070342138379016538987545561, −0.38235683096127059204238013286, 0.38235683096127059204238013286, 1.06070342138379016538987545561, 2.93749784307262355008325067685, 4.26453259039238897813000245729, 5.00461835249379915169013816868, 5.61467486788051575201457821394, 6.76602781915938949456628218229, 7.36752038700786897524586130613, 8.300242219905904869414763619504, 9.613832720862587536655758065324

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