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Properties

Label 2-1176-168.83-c0-0-2
Degree $2$
Conductor $1176$
Sign $0.686 - 0.727i$
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  + i·2-s + (−0.382 − 0.923i)3-s − 4-s + (0.923 − 0.382i)6-s − i·8-s + (−0.707 + 0.707i)9-s + 1.41i·11-s + (0.382 + 0.923i)12-s + 16-s + 1.84·17-s + (−0.707 − 0.707i)18-s − 0.765i·19-s − 1.41·22-s + (−0.923 + 0.382i)24-s + 25-s + ⋯
L(s)  = 1  + i·2-s + (−0.382 − 0.923i)3-s − 4-s + (0.923 − 0.382i)6-s − i·8-s + (−0.707 + 0.707i)9-s + 1.41i·11-s + (0.382 + 0.923i)12-s + 16-s + 1.84·17-s + (−0.707 − 0.707i)18-s − 0.765i·19-s − 1.41·22-s + (−0.923 + 0.382i)24-s + 25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.686 - 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.686 - 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $0.686 - 0.727i$
Analytic conductor: \(0.586900\)
Root analytic conductor: \(0.766094\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (587, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1176,\ (\ :0),\ 0.686 - 0.727i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8465782325\)
\(L(\frac12)\) \(\approx\) \(0.8465782325\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (0.382 + 0.923i)T \)
7 \( 1 \)
good5 \( 1 - T^{2} \)
11 \( 1 - 1.41iT - T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 - 1.84T + T^{2} \)
19 \( 1 + 0.765iT - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - 0.765T + T^{2} \)
43 \( 1 - 1.41T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + 1.84T + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - 1.84iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + 1.84T + T^{2} \)
89 \( 1 + 0.765T + T^{2} \)
97 \( 1 + 0.765iT - T^{2} \)
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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

−9.896906145654032338593617260804, −9.116878782938865551220019950852, −8.110959340720593365131808360264, −7.40141990678705248331490640053, −6.97030428668714674279955564691, −5.95856814499056498277123330053, −5.22550219723157196717573724015, −4.35605964778774852663045486690, −2.84770205761307243537683748443, −1.23011262548324080849652629784, 1.03987638749567628237038698788, 2.96707213638899432864668891032, 3.51064068246838831104142818104, 4.51731487582155330323304320535, 5.56642548281221562991015248327, 5.99589292073494364701541586702, 7.72659103308393380160160938683, 8.544037356733655976761968589950, 9.283245595939622142531983011598, 10.01739006750058217778103615874

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