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Properties

Label 2-229320-1.1-c1-0-17
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 + 2·17-s − 4·19-s + 25-s − 6·29-s + 8·31-s − 10·37-s − 6·41-s − 4·43-s − 8·47-s − 6·53-s + 4·55-s + 12·59-s + 2·61-s − 65-s − 12·67-s + 8·71-s − 10·73-s + 8·79-s + 4·83-s + 2·85-s − 6·89-s − 4·95-s − 18·97-s + 101-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.20·11-s − 0.277·13-s + 0.485·17-s − 0.917·19-s + 1/5·25-s − 1.11·29-s + 1.43·31-s − 1.64·37-s − 0.937·41-s − 0.609·43-s − 1.16·47-s − 0.824·53-s + 0.539·55-s + 1.56·59-s + 0.256·61-s − 0.124·65-s − 1.46·67-s + 0.949·71-s − 1.17·73-s + 0.900·79-s + 0.439·83-s + 0.216·85-s − 0.635·89-s − 0.410·95-s − 1.82·97-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

\(L(1)\) \(\approx\) \(2.009709985\)
\(L(\frac12)\) \(\approx\) \(2.009709985\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−12.94526510091259, −12.40426657111484, −12.03317354261243, −11.57067332876008, −11.16177437644524, −10.52372692843308, −10.03475414721546, −9.781905001077221, −9.196741588698912, −8.660066285202222, −8.380164859020872, −7.762380707183092, −7.087409582774257, −6.659459528407252, −6.376617819919307, −5.726760156819718, −5.194948647482585, −4.713076786317335, −4.101062754032182, −3.568214064216979, −3.095165782279244, −2.332230693968995, −1.690244677648004, −1.358214632588238, −0.3818288099820976, 0.3818288099820976, 1.358214632588238, 1.690244677648004, 2.332230693968995, 3.095165782279244, 3.568214064216979, 4.101062754032182, 4.713076786317335, 5.194948647482585, 5.726760156819718, 6.376617819919307, 6.659459528407252, 7.087409582774257, 7.762380707183092, 8.380164859020872, 8.660066285202222, 9.196741588698912, 9.781905001077221, 10.03475414721546, 10.52372692843308, 11.16177437644524, 11.57067332876008, 12.03317354261243, 12.40426657111484, 12.94526510091259

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