Properties

Label 2-229320-1.1-c1-0-120
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 $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 4·11-s − 13-s − 6·17-s + 8·23-s + 25-s − 6·29-s + 4·31-s − 2·37-s − 10·41-s − 4·43-s − 8·47-s + 2·53-s + 4·55-s + 12·59-s + 2·61-s + 65-s − 16·67-s + 8·71-s + 6·73-s − 16·79-s + 4·83-s + 6·85-s − 2·89-s − 2·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.20·11-s − 0.277·13-s − 1.45·17-s + 1.66·23-s + 1/5·25-s − 1.11·29-s + 0.718·31-s − 0.328·37-s − 1.56·41-s − 0.609·43-s − 1.16·47-s + 0.274·53-s + 0.539·55-s + 1.56·59-s + 0.256·61-s + 0.124·65-s − 1.95·67-s + 0.949·71-s + 0.702·73-s − 1.80·79-s + 0.439·83-s + 0.650·85-s − 0.211·89-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-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: \(2\)
Selberg data: \((2,\ 229320,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−13.44009678283794, −12.95583836737886, −12.65636509821119, −11.80262551790354, −11.60989335126266, −11.06811232833462, −10.65775652530524, −10.21365454470135, −9.717543551249568, −9.074329493595341, −8.683434614363055, −8.279301690802333, −7.700627651125801, −7.271894085141727, −6.598126405810570, −6.576757909573541, −5.480568510359264, −5.141983114884470, −4.831620289279219, −4.110732224854451, −3.610363629421629, −2.858871200453694, −2.593853163042306, −1.829541190186967, −1.151484402740541, 0, 0, 1.151484402740541, 1.829541190186967, 2.593853163042306, 2.858871200453694, 3.610363629421629, 4.110732224854451, 4.831620289279219, 5.141983114884470, 5.480568510359264, 6.576757909573541, 6.598126405810570, 7.271894085141727, 7.700627651125801, 8.279301690802333, 8.683434614363055, 9.074329493595341, 9.717543551249568, 10.21365454470135, 10.65775652530524, 11.06811232833462, 11.60989335126266, 11.80262551790354, 12.65636509821119, 12.95583836737886, 13.44009678283794

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