Properties

Label 2-229320-1.1-c1-0-35
Degree $2$
Conductor $229320$
Sign $1$
Analytic cond. $1831.12$
Root an. cond. $42.7916$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 4·11-s + 13-s − 3·17-s + 6·19-s + 9·23-s + 25-s + 6·31-s − 6·37-s + 5·41-s + 6·43-s + 3·53-s + 4·55-s − 13·59-s − 5·61-s − 65-s + 5·67-s + 71-s − 7·73-s + 6·83-s + 3·85-s + 14·89-s − 6·95-s − 97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.20·11-s + 0.277·13-s − 0.727·17-s + 1.37·19-s + 1.87·23-s + 1/5·25-s + 1.07·31-s − 0.986·37-s + 0.780·41-s + 0.914·43-s + 0.412·53-s + 0.539·55-s − 1.69·59-s − 0.640·61-s − 0.124·65-s + 0.610·67-s + 0.118·71-s − 0.819·73-s + 0.658·83-s + 0.325·85-s + 1.48·89-s − 0.615·95-s − 0.101·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(229320\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(1831.12\)
Root analytic conductor: \(42.7916\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{229320} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 229320,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.277212952\)
\(L(\frac12)\) \(\approx\) \(2.277212952\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + T \)
7 \( 1 \)
13 \( 1 - T \)
good11 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 9 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + 13 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 - T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−12.93166480183657, −12.52858453701163, −11.95474147364244, −11.57000225432938, −10.96713065805389, −10.69041959054609, −10.32477442619527, −9.525954619437825, −9.209670051023354, −8.738615154858953, −8.160739104655318, −7.693428335582388, −7.308391481874579, −6.851122149167119, −6.236305015465267, −5.650221414099671, −5.095116181621246, −4.779812253551495, −4.206699708004485, −3.453972318180022, −2.941418444708217, −2.655800042797283, −1.784696295032790, −1.006528641963336, −0.4872352633923730, 0.4872352633923730, 1.006528641963336, 1.784696295032790, 2.655800042797283, 2.941418444708217, 3.453972318180022, 4.206699708004485, 4.779812253551495, 5.095116181621246, 5.650221414099671, 6.236305015465267, 6.851122149167119, 7.308391481874579, 7.693428335582388, 8.160739104655318, 8.738615154858953, 9.209670051023354, 9.525954619437825, 10.32477442619527, 10.69041959054609, 10.96713065805389, 11.57000225432938, 11.95474147364244, 12.52858453701163, 12.93166480183657

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