Properties

Degree $2$
Conductor $1176$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 0.585·5-s + 9-s − 4.82·11-s + 4.24·13-s − 0.585·15-s + 4.58·17-s + 1.17·19-s + 0.828·23-s − 4.65·25-s − 27-s − 2.82·29-s + 2.82·31-s + 4.82·33-s + 9.65·37-s − 4.24·39-s + 1.75·41-s + 11.3·43-s + 0.585·45-s + 12.4·47-s − 4.58·51-s − 2·53-s − 2.82·55-s − 1.17·57-s − 8.48·59-s − 3.07·61-s + 2.48·65-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.261·5-s + 0.333·9-s − 1.45·11-s + 1.17·13-s − 0.151·15-s + 1.11·17-s + 0.268·19-s + 0.172·23-s − 0.931·25-s − 0.192·27-s − 0.525·29-s + 0.508·31-s + 0.840·33-s + 1.58·37-s − 0.679·39-s + 0.274·41-s + 1.72·43-s + 0.0873·45-s + 1.82·47-s − 0.642·51-s − 0.274·53-s − 0.381·55-s − 0.155·57-s − 1.10·59-s − 0.393·61-s + 0.308·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{1176} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1176,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.380729719\)
\(L(\frac12)\) \(\approx\) \(1.380729719\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 \)
good5 \( 1 - 0.585T + 5T^{2} \)
11 \( 1 + 4.82T + 11T^{2} \)
13 \( 1 - 4.24T + 13T^{2} \)
17 \( 1 - 4.58T + 17T^{2} \)
19 \( 1 - 1.17T + 19T^{2} \)
23 \( 1 - 0.828T + 23T^{2} \)
29 \( 1 + 2.82T + 29T^{2} \)
31 \( 1 - 2.82T + 31T^{2} \)
37 \( 1 - 9.65T + 37T^{2} \)
41 \( 1 - 1.75T + 41T^{2} \)
43 \( 1 - 11.3T + 43T^{2} \)
47 \( 1 - 12.4T + 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 + 8.48T + 59T^{2} \)
61 \( 1 + 3.07T + 61T^{2} \)
67 \( 1 + 11.3T + 67T^{2} \)
71 \( 1 - 6.48T + 71T^{2} \)
73 \( 1 - 16.2T + 73T^{2} \)
79 \( 1 - 2.34T + 79T^{2} \)
83 \( 1 + 4T + 83T^{2} \)
89 \( 1 - 14.2T + 89T^{2} \)
97 \( 1 - 8.24T + 97T^{2} \)
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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

−9.874999623649005635059614898504, −9.070436876700283002549273606440, −7.87350155908191225899270462910, −7.52823551613894385119104245611, −6.03441000081613018251875105271, −5.79067913139686146174150171718, −4.73148377689358237311182151985, −3.61151833613487952017859474691, −2.42875513623203969810029701258, −0.932550693858387367066718846611, 0.932550693858387367066718846611, 2.42875513623203969810029701258, 3.61151833613487952017859474691, 4.73148377689358237311182151985, 5.79067913139686146174150171718, 6.03441000081613018251875105271, 7.52823551613894385119104245611, 7.87350155908191225899270462910, 9.070436876700283002549273606440, 9.874999623649005635059614898504

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