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Properties

Label 2-1176-21.20-c1-0-38
Degree $2$
Conductor $1176$
Sign $0.114 + 0.993i$
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  + (1.48 − 0.884i)3-s + 0.992·5-s + (1.43 − 2.63i)9-s − 2.50i·11-s − 6.74i·13-s + (1.47 − 0.877i)15-s − 3.75·17-s + 2.67i·19-s − 1.74i·23-s − 4.01·25-s + (−0.190 − 5.19i)27-s + 6.69i·29-s − 2.82i·31-s + (−2.21 − 3.73i)33-s − 3.28·37-s + ⋯
L(s)  = 1  + (0.859 − 0.510i)3-s + 0.443·5-s + (0.478 − 0.877i)9-s − 0.756i·11-s − 1.87i·13-s + (0.381 − 0.226i)15-s − 0.911·17-s + 0.612i·19-s − 0.364i·23-s − 0.803·25-s + (−0.0366 − 0.999i)27-s + 1.24i·29-s − 0.507i·31-s + (−0.386 − 0.650i)33-s − 0.539·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.114 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.114 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $0.114 + 0.993i$
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: \((2,\ 1176,\ (\ :1/2),\ 0.114 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.234497611\)
\(L(\frac12)\) \(\approx\) \(2.234497611\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.48 + 0.884i)T \)
7 \( 1 \)
good5 \( 1 - 0.992T + 5T^{2} \)
11 \( 1 + 2.50iT - 11T^{2} \)
13 \( 1 + 6.74iT - 13T^{2} \)
17 \( 1 + 3.75T + 17T^{2} \)
19 \( 1 - 2.67iT - 19T^{2} \)
23 \( 1 + 1.74iT - 23T^{2} \)
29 \( 1 - 6.69iT - 29T^{2} \)
31 \( 1 + 2.82iT - 31T^{2} \)
37 \( 1 + 3.28T + 37T^{2} \)
41 \( 1 - 9.02T + 41T^{2} \)
43 \( 1 - 10.6T + 43T^{2} \)
47 \( 1 - 10.5T + 47T^{2} \)
53 \( 1 - 10.2iT - 53T^{2} \)
59 \( 1 - 4.64T + 59T^{2} \)
61 \( 1 + 8.05iT - 61T^{2} \)
67 \( 1 + 6.69T + 67T^{2} \)
71 \( 1 + 4.56iT - 71T^{2} \)
73 \( 1 - 3.23iT - 73T^{2} \)
79 \( 1 + 14.4T + 79T^{2} \)
83 \( 1 - 16.8T + 83T^{2} \)
89 \( 1 + 3.17T + 89T^{2} \)
97 \( 1 + 0.989iT - 97T^{2} \)
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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

−9.384384568244965374219052087383, −8.750882495401299434336474468616, −7.941891803995345707602649279261, −7.33258592142726528304347310476, −6.13363542528387610860929910869, −5.60401587811545343574184893769, −4.14447945244788738326221063814, −3.13979667176962058958629142087, −2.30356110565696414479569139242, −0.874294284007256595537399700086, 1.88133494075344524241816345993, 2.51801416501138672679477822540, 4.12999367129053324957811455822, 4.40044746901993852954498595219, 5.69647388092264375035525511282, 6.84045834667892742814105013993, 7.43713736210778902843428255438, 8.575013721919964448384375788830, 9.318369458758999465391890899777, 9.611651189816291148038535751235

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