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Properties

Label 2-1176-21.20-c1-0-26
Degree $2$
Conductor $1176$
Sign $0.707 + 0.706i$
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  + (−0.615 + 1.61i)3-s + 1.31·5-s + (−2.24 − 1.99i)9-s − 4.97i·11-s + 0.733i·13-s + (−0.806 + 2.12i)15-s + 0.728·17-s − 6.94i·19-s − 7.92i·23-s − 3.27·25-s + (4.60 − 2.40i)27-s + 4.62i·29-s − 1.80i·31-s + (8.06 + 3.06i)33-s − 5.22·37-s + ⋯
L(s)  = 1  + (−0.355 + 0.934i)3-s + 0.586·5-s + (−0.747 − 0.663i)9-s − 1.50i·11-s + 0.203i·13-s + (−0.208 + 0.548i)15-s + 0.176·17-s − 1.59i·19-s − 1.65i·23-s − 0.655·25-s + (0.886 − 0.463i)27-s + 0.859i·29-s − 0.323i·31-s + (1.40 + 0.533i)33-s − 0.858·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.706i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $0.707 + 0.706i$
Analytic conductor: \(9.39040\)
Root analytic conductor: \(3.06437\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1176,\ (\ :1/2),\ 0.707 + 0.706i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.265183862\)
\(L(\frac12)\) \(\approx\) \(1.265183862\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.615 - 1.61i)T \)
7 \( 1 \)
good5 \( 1 - 1.31T + 5T^{2} \)
11 \( 1 + 4.97iT - 11T^{2} \)
13 \( 1 - 0.733iT - 13T^{2} \)
17 \( 1 - 0.728T + 17T^{2} \)
19 \( 1 + 6.94iT - 19T^{2} \)
23 \( 1 + 7.92iT - 23T^{2} \)
29 \( 1 - 4.62iT - 29T^{2} \)
31 \( 1 + 1.80iT - 31T^{2} \)
37 \( 1 + 5.22T + 37T^{2} \)
41 \( 1 - 7.83T + 41T^{2} \)
43 \( 1 + 1.27T + 43T^{2} \)
47 \( 1 + 4.24T + 47T^{2} \)
53 \( 1 + 14.4iT - 53T^{2} \)
59 \( 1 - 9.35T + 59T^{2} \)
61 \( 1 - 13.5iT - 61T^{2} \)
67 \( 1 + 7.54T + 67T^{2} \)
71 \( 1 - 6.92iT - 71T^{2} \)
73 \( 1 - 3.79iT - 73T^{2} \)
79 \( 1 - 13.5T + 79T^{2} \)
83 \( 1 - 6.58T + 83T^{2} \)
89 \( 1 + 4.37T + 89T^{2} \)
97 \( 1 + 15.1iT - 97T^{2} \)
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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

−9.727859334974754533291453183156, −8.835883521498711220125778704102, −8.463446329954551382253939241869, −6.95155392287558638247364852502, −6.15037212803954774626453110822, −5.42349048592942648857370428378, −4.56601652999445717954402923658, −3.48886481495881104249463871695, −2.52401006099284430444863526310, −0.57774899628747049866871430534, 1.49263914702006069986057795632, 2.19931903900285164811193310028, 3.66355958363186160775104350300, 4.97878412782425264183217798541, 5.78444728551773417151834118674, 6.46482285887177871930680281665, 7.66240483673757980735948631227, 7.74715711380498342946676004215, 9.180389745674236187589354256033, 9.890864026064374190665990609022

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