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Properties

Label 2-1176-168.83-c0-0-0
Degree $2$
Conductor $1176$
Sign $0.972 + 0.233i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 + 0.233i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 + 0.233i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $0.972 + 0.233i$
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: \((2,\ 1176,\ (\ :0),\ 0.972 + 0.233i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6167038388\)
\(L(\frac12)\) \(\approx\) \(0.6167038388\)
\(L(1)\) 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

−10.13251161477612807456998133714, −9.481960528940733296577058501925, −8.345996007674178196962104412818, −7.50406665184748683711325140966, −6.55362214039505143103951916654, −5.52506504957566901299281077819, −4.68766926356519471978861630510, −3.92261642676476347253609707041, −2.36671393450021905834027270764, −1.42248574590905542223413492015, 0.68543810693066189764620871188, 3.11466019514280881075898962050, 4.30461392333731297353600600417, 5.01948765119054007576279930394, 5.86163576153203248167134296071, 6.59849244482153338889583966864, 7.23399881269550642705127092974, 8.504311286650031332545716832415, 9.011445601301270152857969144945, 9.860305991335665782878663345390

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