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Properties

Label 2-1176-56.27-c1-0-78
Degree $2$
Conductor $1176$
Sign $-0.999 + 0.0247i$
Analytic cond. $9.39040$
Root an. cond. $3.06437$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.35 − 0.409i)2-s + i·3-s + (1.66 − 1.10i)4-s − 2.50·5-s + (0.409 + 1.35i)6-s + (1.79 − 2.18i)8-s − 9-s + (−3.38 + 1.02i)10-s − 5.67·11-s + (1.10 + 1.66i)12-s − 5.31·13-s − 2.50i·15-s + (1.53 − 3.69i)16-s − 0.454i·17-s + (−1.35 + 0.409i)18-s + 3.69i·19-s + ⋯
L(s)  = 1  + (0.957 − 0.289i)2-s + 0.577i·3-s + (0.832 − 0.554i)4-s − 1.11·5-s + (0.167 + 0.552i)6-s + (0.635 − 0.771i)8-s − 0.333·9-s + (−1.07 + 0.324i)10-s − 1.71·11-s + (0.320 + 0.480i)12-s − 1.47·13-s − 0.646i·15-s + (0.384 − 0.922i)16-s − 0.110i·17-s + (−0.319 + 0.0965i)18-s + 0.847i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0247i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0247i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $-0.999 + 0.0247i$
Analytic conductor: \(9.39040\)
Root analytic conductor: \(3.06437\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (979, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1176,\ (\ :1/2),\ -0.999 + 0.0247i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1362892118\)
\(L(\frac12)\) \(\approx\) \(0.1362892118\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.35 + 0.409i)T \)
3 \( 1 - iT \)
7 \( 1 \)
good5 \( 1 + 2.50T + 5T^{2} \)
11 \( 1 + 5.67T + 11T^{2} \)
13 \( 1 + 5.31T + 13T^{2} \)
17 \( 1 + 0.454iT - 17T^{2} \)
19 \( 1 - 3.69iT - 19T^{2} \)
23 \( 1 + 5.12iT - 23T^{2} \)
29 \( 1 - 2.57iT - 29T^{2} \)
31 \( 1 + 6.00T + 31T^{2} \)
37 \( 1 - 9.01iT - 37T^{2} \)
41 \( 1 + 4.65iT - 41T^{2} \)
43 \( 1 - 3.66T + 43T^{2} \)
47 \( 1 - 0.957T + 47T^{2} \)
53 \( 1 + 6.24iT - 53T^{2} \)
59 \( 1 + 10.1iT - 59T^{2} \)
61 \( 1 - 5.00T + 61T^{2} \)
67 \( 1 + 9.30T + 67T^{2} \)
71 \( 1 - 7.35iT - 71T^{2} \)
73 \( 1 - 6.85iT - 73T^{2} \)
79 \( 1 - 8.91iT - 79T^{2} \)
83 \( 1 + 1.96iT - 83T^{2} \)
89 \( 1 + 6.82iT - 89T^{2} \)
97 \( 1 + 3.71iT - 97T^{2} \)
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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

−9.755751961920723700150581539342, −8.333854074288789301559372617144, −7.65399763303151078497790699388, −6.90627338220294639528046613406, −5.54540969948244288693010313967, −4.95788405059916545540716677948, −4.17112443103935997645664093448, −3.18909414242931029412059621872, −2.33246383881114962340853136439, −0.03781899413295415584395633584, 2.26150164460901292615201124232, 3.04448246965385196599862403532, 4.21238618566345381605548591766, 5.10004619360485878469197068672, 5.79386165044088812093343240100, 7.17564771903040129899513015998, 7.52819210038636719672005092183, 7.980550999058790706728044684643, 9.220659775972575471520119392205, 10.51728358628270417937249523537

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