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Properties

Label 2-229320-1.1-c1-0-73
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 + 5·11-s − 13-s + 7·17-s + 6·19-s − 3·23-s + 25-s + 6·29-s + 4·31-s + 7·37-s + 3·41-s − 10·47-s + 6·53-s + 5·55-s − 9·59-s − 2·61-s − 65-s − 9·67-s − 16·71-s − 73-s + 3·79-s + 16·83-s + 7·85-s + 5·89-s + 6·95-s + 17·97-s + 101-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.50·11-s − 0.277·13-s + 1.69·17-s + 1.37·19-s − 0.625·23-s + 1/5·25-s + 1.11·29-s + 0.718·31-s + 1.15·37-s + 0.468·41-s − 1.45·47-s + 0.824·53-s + 0.674·55-s − 1.17·59-s − 0.256·61-s − 0.124·65-s − 1.09·67-s − 1.89·71-s − 0.117·73-s + 0.337·79-s + 1.75·83-s + 0.759·85-s + 0.529·89-s + 0.615·95-s + 1.72·97-s + 0.0995·101-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 229320,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.718280134\)
\(L(\frac12)\) \(\approx\) \(4.718280134\)
\(L(\frac{3}{2})\) 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 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 9 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + T + p T^{2} \)
79 \( 1 - 3 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 5 T + p T^{2} \)
97 \( 1 - 17 T + p T^{2} \)
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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

−13.01784731005985, −12.15804656465741, −11.98525768957029, −11.83558512798603, −11.15700265226624, −10.45384408411951, −10.02915504659856, −9.725176079501629, −9.261507482914982, −8.835287029369890, −8.174858174858743, −7.558289881971153, −7.478763948654430, −6.537168328324608, −6.255369499898836, −5.840099369108674, −5.192840895268310, −4.681220578603924, −4.176726676424181, −3.425077677739743, −3.130393429650003, −2.485763742015473, −1.586783026618769, −1.207475783302115, −0.6677156874795741, 0.6677156874795741, 1.207475783302115, 1.586783026618769, 2.485763742015473, 3.130393429650003, 3.425077677739743, 4.176726676424181, 4.681220578603924, 5.192840895268310, 5.840099369108674, 6.255369499898836, 6.537168328324608, 7.478763948654430, 7.558289881971153, 8.174858174858743, 8.835287029369890, 9.261507482914982, 9.725176079501629, 10.02915504659856, 10.45384408411951, 11.15700265226624, 11.83558512798603, 11.98525768957029, 12.15804656465741, 13.01784731005985

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