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Properties

Label 2-1176-168.83-c0-0-4
Degree $2$
Conductor $1176$
Sign $0.852 + 0.522i$
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 + (0.923 + 0.382i)3-s − 4-s + (0.382 − 0.923i)6-s + i·8-s + (0.707 + 0.707i)9-s + 1.41i·11-s + (−0.923 − 0.382i)12-s + 16-s + 0.765·17-s + (0.707 − 0.707i)18-s − 1.84i·19-s + 1.41·22-s + (−0.382 + 0.923i)24-s + 25-s + ⋯
L(s)  = 1  − i·2-s + (0.923 + 0.382i)3-s − 4-s + (0.382 − 0.923i)6-s + i·8-s + (0.707 + 0.707i)9-s + 1.41i·11-s + (−0.923 − 0.382i)12-s + 16-s + 0.765·17-s + (0.707 − 0.707i)18-s − 1.84i·19-s + 1.41·22-s + (−0.382 + 0.923i)24-s + 25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.852 + 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.852 + 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $0.852 + 0.522i$
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: \((2,\ 1176,\ (\ :0),\ 0.852 + 0.522i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.318620776\)
\(L(\frac12)\) \(\approx\) \(1.318620776\)
\(L(1)\) 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 \( 1 + (-0.923 - 0.382i)T \)
7 \( 1 \)
good5 \( 1 - T^{2} \)
11 \( 1 - 1.41iT - T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 - 0.765T + T^{2} \)
19 \( 1 + 1.84iT - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + 1.84T + T^{2} \)
43 \( 1 + 1.41T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + 0.765T + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + 0.765iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + 0.765T + T^{2} \)
89 \( 1 - 1.84T + T^{2} \)
97 \( 1 + 1.84iT - T^{2} \)
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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

−9.965604428576895854815377698533, −9.201237708849478591865727916824, −8.609571236159192419210060906294, −7.63441804529017202481164675502, −6.81270452261645877697044157186, −4.99917799138492074845388914767, −4.70100027277823120381219535865, −3.49954813567124399038427907134, −2.68019146047517212366029777280, −1.65245981197382830286864866599, 1.34200109399420800243624913450, 3.21032566095932837888166514836, 3.74786251263512593717778077188, 5.11269002888946696746416179486, 6.04032363528625993169857132171, 6.75165630839988987386016898746, 7.77332648177743860089629296787, 8.298313523513676385023077091965, 8.843257333481443878611605500326, 9.840773098565069496397581658567

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