Properties

Label 2-229320-1.1-c1-0-67
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 $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 5·11-s + 13-s − 2·17-s + 6·19-s − 9·23-s + 25-s − 6·29-s + 5·31-s + 37-s + 3·41-s + 6·43-s − 7·47-s + 12·53-s + 5·55-s − 12·59-s − 61-s − 65-s − 9·67-s − 8·71-s − 5·73-s − 3·79-s − 2·83-s + 2·85-s + 14·89-s − 6·95-s + 13·97-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.50·11-s + 0.277·13-s − 0.485·17-s + 1.37·19-s − 1.87·23-s + 1/5·25-s − 1.11·29-s + 0.898·31-s + 0.164·37-s + 0.468·41-s + 0.914·43-s − 1.02·47-s + 1.64·53-s + 0.674·55-s − 1.56·59-s − 0.128·61-s − 0.124·65-s − 1.09·67-s − 0.949·71-s − 0.585·73-s − 0.337·79-s − 0.219·83-s + 0.216·85-s + 1.48·89-s − 0.615·95-s + 1.31·97-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: \(1\)
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 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 + 9 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 5 T + p T^{2} \)
79 \( 1 + 3 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.29220276019066, −12.73624213920924, −12.14785293915758, −11.73974081716019, −11.45289677987066, −10.72440846803442, −10.45120158886121, −9.958004379163455, −9.485249050524419, −8.892741679375447, −8.427240300281095, −7.811897068130376, −7.562305646423769, −7.274752812439560, −6.350629710386461, −5.885052543610335, −5.603196383469486, −4.788149104443390, −4.538986418785105, −3.795997885831681, −3.320788750892304, −2.708657888215268, −2.199847336479129, −1.513706956641004, −0.6419307187487250, 0, 0.6419307187487250, 1.513706956641004, 2.199847336479129, 2.708657888215268, 3.320788750892304, 3.795997885831681, 4.538986418785105, 4.788149104443390, 5.603196383469486, 5.885052543610335, 6.350629710386461, 7.274752812439560, 7.562305646423769, 7.811897068130376, 8.427240300281095, 8.892741679375447, 9.485249050524419, 9.958004379163455, 10.45120158886121, 10.72440846803442, 11.45289677987066, 11.73974081716019, 12.14785293915758, 12.73624213920924, 13.29220276019066

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