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Properties

Label 2-1176-56.27-c1-0-15
Degree $2$
Conductor $1176$
Sign $-0.220 - 0.975i$
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  + (−1.04 − 0.948i)2-s + i·3-s + (0.200 + 1.98i)4-s + 2.88·5-s + (0.948 − 1.04i)6-s + (1.67 − 2.27i)8-s − 9-s + (−3.02 − 2.73i)10-s − 5.82·11-s + (−1.98 + 0.200i)12-s − 1.04·13-s + 2.88i·15-s + (−3.91 + 0.796i)16-s + 6.82i·17-s + (1.04 + 0.948i)18-s − 0.681i·19-s + ⋯
L(s)  = 1  + (−0.741 − 0.670i)2-s + 0.577i·3-s + (0.100 + 0.994i)4-s + 1.28·5-s + (0.387 − 0.428i)6-s + (0.593 − 0.805i)8-s − 0.333·9-s + (−0.956 − 0.864i)10-s − 1.75·11-s + (−0.574 + 0.0577i)12-s − 0.290·13-s + 0.744i·15-s + (−0.979 + 0.199i)16-s + 1.65i·17-s + (0.247 + 0.223i)18-s − 0.156i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.220 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.220 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $-0.220 - 0.975i$
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: \((2,\ 1176,\ (\ :1/2),\ -0.220 - 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7962062731\)
\(L(\frac12)\) \(\approx\) \(0.7962062731\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.04 + 0.948i)T \)
3 \( 1 - iT \)
7 \( 1 \)
good5 \( 1 - 2.88T + 5T^{2} \)
11 \( 1 + 5.82T + 11T^{2} \)
13 \( 1 + 1.04T + 13T^{2} \)
17 \( 1 - 6.82iT - 17T^{2} \)
19 \( 1 + 0.681iT - 19T^{2} \)
23 \( 1 - 2.14iT - 23T^{2} \)
29 \( 1 - 6.61iT - 29T^{2} \)
31 \( 1 + 3.83T + 31T^{2} \)
37 \( 1 - 2.38iT - 37T^{2} \)
41 \( 1 - 1.19iT - 41T^{2} \)
43 \( 1 - 1.34T + 43T^{2} \)
47 \( 1 + 11.0T + 47T^{2} \)
53 \( 1 - 8.07iT - 53T^{2} \)
59 \( 1 + 7.87iT - 59T^{2} \)
61 \( 1 - 3.26T + 61T^{2} \)
67 \( 1 - 13.3T + 67T^{2} \)
71 \( 1 - 1.08iT - 71T^{2} \)
73 \( 1 + 5.64iT - 73T^{2} \)
79 \( 1 - 12.6iT - 79T^{2} \)
83 \( 1 + 0.482iT - 83T^{2} \)
89 \( 1 - 12.3iT - 89T^{2} \)
97 \( 1 - 3.63iT - 97T^{2} \)
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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

−9.946153303875996948040579890238, −9.518971643979497224647211483356, −8.500474880229682342126727834432, −7.908488130033462729615753447819, −6.74405560442342011698261643735, −5.66379903731968433117927097877, −4.91348365921320012357705904214, −3.57793768332163276319169316279, −2.58454038139381017088635227108, −1.67714278389013110526048108609, 0.40402690490794365808913882045, 2.02567720499037652003676567933, 2.67009212178742872370074681598, 4.94320077106570934696461242706, 5.45587426656853331548082021545, 6.25948649098126696825419703705, 7.17664088467779524721006443158, 7.80051059209762665418311986639, 8.654450049693321807651346591944, 9.649571716970131207818879717138

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