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Properties

Label 2-1176-1176.389-c0-0-1
Degree $2$
Conductor $1176$
Sign $-0.814 - 0.580i$
Analytic cond. $0.586900$
Root an. cond. $0.766094$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.365 − 0.930i)2-s + (0.0747 − 0.997i)3-s + (−0.733 − 0.680i)4-s + (−1.63 − 1.11i)5-s + (−0.900 − 0.433i)6-s + (0.826 − 0.563i)7-s + (−0.900 + 0.433i)8-s + (−0.988 − 0.149i)9-s + (−1.63 + 1.11i)10-s + (1.44 − 0.218i)11-s + (−0.733 + 0.680i)12-s + (−0.222 − 0.974i)14-s + (−1.23 + 1.54i)15-s + (0.0747 + 0.997i)16-s + (−0.5 + 0.866i)18-s + ⋯
L(s)  = 1  + (0.365 − 0.930i)2-s + (0.0747 − 0.997i)3-s + (−0.733 − 0.680i)4-s + (−1.63 − 1.11i)5-s + (−0.900 − 0.433i)6-s + (0.826 − 0.563i)7-s + (−0.900 + 0.433i)8-s + (−0.988 − 0.149i)9-s + (−1.63 + 1.11i)10-s + (1.44 − 0.218i)11-s + (−0.733 + 0.680i)12-s + (−0.222 − 0.974i)14-s + (−1.23 + 1.54i)15-s + (0.0747 + 0.997i)16-s + (−0.5 + 0.866i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.814 - 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.814 - 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $-0.814 - 0.580i$
Analytic conductor: \(0.586900\)
Root analytic conductor: \(0.766094\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (389, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1176,\ (\ :0),\ -0.814 - 0.580i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9178600188\)
\(L(\frac12)\) \(\approx\) \(0.9178600188\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.365 + 0.930i)T \)
3 \( 1 + (-0.0747 + 0.997i)T \)
7 \( 1 + (-0.826 + 0.563i)T \)
good5 \( 1 + (1.63 + 1.11i)T + (0.365 + 0.930i)T^{2} \)
11 \( 1 + (-1.44 + 0.218i)T + (0.955 - 0.294i)T^{2} \)
13 \( 1 + (0.222 - 0.974i)T^{2} \)
17 \( 1 + (-0.826 - 0.563i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.826 + 0.563i)T^{2} \)
29 \( 1 + (0.162 + 0.712i)T + (-0.900 + 0.433i)T^{2} \)
31 \( 1 + (0.0747 - 0.129i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.0747 + 0.997i)T^{2} \)
41 \( 1 + (-0.623 + 0.781i)T^{2} \)
43 \( 1 + (-0.623 - 0.781i)T^{2} \)
47 \( 1 + (0.733 + 0.680i)T^{2} \)
53 \( 1 + (-0.733 - 0.680i)T + (0.0747 + 0.997i)T^{2} \)
59 \( 1 + (0.826 - 0.563i)T + (0.365 - 0.930i)T^{2} \)
61 \( 1 + (-0.0747 + 0.997i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.900 + 0.433i)T^{2} \)
73 \( 1 + (0.658 + 1.67i)T + (-0.733 + 0.680i)T^{2} \)
79 \( 1 + (0.826 + 1.43i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.455 + 0.571i)T + (-0.222 - 0.974i)T^{2} \)
89 \( 1 + (-0.955 - 0.294i)T^{2} \)
97 \( 1 - 1.91T + T^{2} \)
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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

−9.178086738204346826133737093553, −8.723939611422145144656595357757, −7.943505100125859610203544986859, −7.22523811416588744357422716277, −5.99889690126097983130988469172, −4.82977488784210721539728087644, −4.14707317234359410581369641576, −3.36828189752933158649202343886, −1.64328853643776430245829809148, −0.801121348777222514376618643136, 2.87315484052320250471011083551, 3.83814356872115677716285344817, 4.26548593249801614551255723624, 5.25893342302764392879322728246, 6.40079644761875697872302585068, 7.14056418447894258601508294085, 8.029264084247829895687374091992, 8.610466682645294554083572671342, 9.389928018235517917117366997218, 10.52711809178378284144852638589

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