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Properties

Label 2-1176-1176.701-c0-0-0
Degree $2$
Conductor $1176$
Sign $0.967 - 0.253i$
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  + (0.222 − 0.974i)2-s + (−0.623 + 0.781i)3-s + (−0.900 − 0.433i)4-s + (0.277 − 0.347i)5-s + (0.623 + 0.781i)6-s + (0.623 + 0.781i)7-s + (−0.623 + 0.781i)8-s + (−0.222 − 0.974i)9-s + (−0.277 − 0.347i)10-s + (−0.400 + 1.75i)11-s + (0.900 − 0.433i)12-s + (0.900 − 0.433i)14-s + (0.0990 + 0.433i)15-s + (0.623 + 0.781i)16-s − 18-s + ⋯
L(s)  = 1  + (0.222 − 0.974i)2-s + (−0.623 + 0.781i)3-s + (−0.900 − 0.433i)4-s + (0.277 − 0.347i)5-s + (0.623 + 0.781i)6-s + (0.623 + 0.781i)7-s + (−0.623 + 0.781i)8-s + (−0.222 − 0.974i)9-s + (−0.277 − 0.347i)10-s + (−0.400 + 1.75i)11-s + (0.900 − 0.433i)12-s + (0.900 − 0.433i)14-s + (0.0990 + 0.433i)15-s + (0.623 + 0.781i)16-s − 18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $0.967 - 0.253i$
Analytic conductor: \(0.586900\)
Root analytic conductor: \(0.766094\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (701, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1176,\ (\ :0),\ 0.967 - 0.253i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8869829398\)
\(L(\frac12)\) \(\approx\) \(0.8869829398\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.222 + 0.974i)T \)
3 \( 1 + (0.623 - 0.781i)T \)
7 \( 1 + (-0.623 - 0.781i)T \)
good5 \( 1 + (-0.277 + 0.347i)T + (-0.222 - 0.974i)T^{2} \)
11 \( 1 + (0.400 - 1.75i)T + (-0.900 - 0.433i)T^{2} \)
13 \( 1 + (0.900 + 0.433i)T^{2} \)
17 \( 1 + (-0.623 + 0.781i)T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-0.623 - 0.781i)T^{2} \)
29 \( 1 + (0.400 - 0.193i)T + (0.623 - 0.781i)T^{2} \)
31 \( 1 - 1.24T + T^{2} \)
37 \( 1 + (-0.623 + 0.781i)T^{2} \)
41 \( 1 + (0.222 + 0.974i)T^{2} \)
43 \( 1 + (0.222 - 0.974i)T^{2} \)
47 \( 1 + (0.900 + 0.433i)T^{2} \)
53 \( 1 + (-1.80 - 0.867i)T + (0.623 + 0.781i)T^{2} \)
59 \( 1 + (1.24 + 1.56i)T + (-0.222 + 0.974i)T^{2} \)
61 \( 1 + (-0.623 + 0.781i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.623 - 0.781i)T^{2} \)
73 \( 1 + (0.277 + 1.21i)T + (-0.900 + 0.433i)T^{2} \)
79 \( 1 - 1.24T + T^{2} \)
83 \( 1 + (0.0990 + 0.433i)T + (-0.900 + 0.433i)T^{2} \)
89 \( 1 + (0.900 - 0.433i)T^{2} \)
97 \( 1 + 1.80T + T^{2} \)
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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

−10.07955633446522352483803790432, −9.406008278898475606384851454045, −8.844448333433572727134553911191, −7.70181080746895159990458588569, −6.32889247775136445794565460835, −5.25478117463158791202229430050, −4.91835830755437532616866468401, −4.06323084783848701375003302259, −2.71120291887870752498793697656, −1.60931141847448354532487306372, 0.867976066873242281925285909623, 2.76287358634860204171039607823, 4.06381108015575622882285095782, 5.14019350162942212114475845034, 5.88228936954331627901823873948, 6.54955376410632655607476375198, 7.35922333264478913022648803286, 8.157027357034902474394544975367, 8.618063089401570790892325584085, 10.05443758037927956287049623967

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