Properties

Label 2-229320-1.1-c1-0-1
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 − 6·17-s + 6·23-s + 25-s − 6·29-s − 6·31-s − 6·37-s + 2·41-s + 4·55-s − 4·59-s − 2·61-s − 65-s + 2·67-s − 8·71-s + 8·73-s + 12·83-s + 6·85-s + 14·89-s + 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 6·115-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.20·11-s + 0.277·13-s − 1.45·17-s + 1.25·23-s + 1/5·25-s − 1.11·29-s − 1.07·31-s − 0.986·37-s + 0.312·41-s + 0.539·55-s − 0.520·59-s − 0.256·61-s − 0.124·65-s + 0.244·67-s − 0.949·71-s + 0.936·73-s + 1.31·83-s + 0.650·85-s + 1.48·89-s + 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 0.559·115-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

\(L(1)\) \(\approx\) \(0.3861171878\)
\(L(\frac12)\) \(\approx\) \(0.3861171878\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−12.95843621063127, −12.58121680234642, −12.02807427939147, −11.41110976365473, −10.94879600205072, −10.80169847205805, −10.32142355154674, −9.561486967363189, −8.987906737415698, −8.915286068426897, −8.158056510352810, −7.717492245442631, −7.273494887424999, −6.818452594131324, −6.289025573153247, −5.626588790198693, −5.113811464239548, −4.798917538723218, −4.089763137049105, −3.576104556232524, −3.052949688319889, −2.371525679511174, −1.935270539531568, −1.105980726465562, −0.1793079756844870, 0.1793079756844870, 1.105980726465562, 1.935270539531568, 2.371525679511174, 3.052949688319889, 3.576104556232524, 4.089763137049105, 4.798917538723218, 5.113811464239548, 5.626588790198693, 6.289025573153247, 6.818452594131324, 7.273494887424999, 7.717492245442631, 8.158056510352810, 8.915286068426897, 8.987906737415698, 9.561486967363189, 10.32142355154674, 10.80169847205805, 10.94879600205072, 11.41110976365473, 12.02807427939147, 12.58121680234642, 12.95843621063127

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