Properties

Degree $2$
Conductor $1134$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 1.44·5-s − 7-s + 8-s − 1.44·10-s + 2·11-s + 4.89·13-s − 14-s + 16-s + 2·17-s + 2.55·19-s − 1.44·20-s + 2·22-s − 23-s − 2.89·25-s + 4.89·26-s − 28-s − 6.89·29-s + 6·31-s + 32-s + 2·34-s + 1.44·35-s + 11.7·37-s + 2.55·38-s − 1.44·40-s + 9.79·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 0.648·5-s − 0.377·7-s + 0.353·8-s − 0.458·10-s + 0.603·11-s + 1.35·13-s − 0.267·14-s + 0.250·16-s + 0.485·17-s + 0.585·19-s − 0.324·20-s + 0.426·22-s − 0.208·23-s − 0.579·25-s + 0.960·26-s − 0.188·28-s − 1.28·29-s + 1.07·31-s + 0.176·32-s + 0.342·34-s + 0.245·35-s + 1.93·37-s + 0.413·38-s − 0.229·40-s + 1.53·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{1134} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.417177244\)
\(L(\frac12)\) \(\approx\) \(2.417177244\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
7 \( 1 + T \)
good5 \( 1 + 1.44T + 5T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 - 4.89T + 13T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 - 2.55T + 19T^{2} \)
23 \( 1 + T + 23T^{2} \)
29 \( 1 + 6.89T + 29T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 - 11.7T + 37T^{2} \)
41 \( 1 - 9.79T + 41T^{2} \)
43 \( 1 - 6.89T + 43T^{2} \)
47 \( 1 - 9.79T + 47T^{2} \)
53 \( 1 + 10.8T + 53T^{2} \)
59 \( 1 + 2T + 59T^{2} \)
61 \( 1 - 6.55T + 61T^{2} \)
67 \( 1 + 12.8T + 67T^{2} \)
71 \( 1 - 0.101T + 71T^{2} \)
73 \( 1 + 6.89T + 73T^{2} \)
79 \( 1 + 1.89T + 79T^{2} \)
83 \( 1 - 2T + 83T^{2} \)
89 \( 1 + 16.8T + 89T^{2} \)
97 \( 1 - 2.89T + 97T^{2} \)
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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

−9.793494786784438501371574931727, −9.046057312394664916492155775652, −7.944597885090500187252565392832, −7.37329037389303157850569248409, −6.16482514106445268229372327869, −5.79017344906158182702940994675, −4.30194463606659594268058444029, −3.81694937894669710576354089081, −2.78007502426693272047252555679, −1.15623785622867158010014689522, 1.15623785622867158010014689522, 2.78007502426693272047252555679, 3.81694937894669710576354089081, 4.30194463606659594268058444029, 5.79017344906158182702940994675, 6.16482514106445268229372327869, 7.37329037389303157850569248409, 7.944597885090500187252565392832, 9.046057312394664916492155775652, 9.793494786784438501371574931727

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