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Properties

Label 2-1176-168.149-c0-0-4
Degree $2$
Conductor $1176$
Sign $0.925 + 0.378i$
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  + (0.866 + 0.5i)2-s + (0.258 − 0.965i)3-s + (0.499 + 0.866i)4-s + (0.707 − 1.22i)5-s + (0.707 − 0.707i)6-s + 0.999i·8-s + (−0.866 − 0.499i)9-s + (1.22 − 0.707i)10-s + (0.965 − 0.258i)12-s + 1.41i·13-s + (−0.999 − i)15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)18-s + (−1.22 − 0.707i)19-s + 1.41·20-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.258 − 0.965i)3-s + (0.499 + 0.866i)4-s + (0.707 − 1.22i)5-s + (0.707 − 0.707i)6-s + 0.999i·8-s + (−0.866 − 0.499i)9-s + (1.22 − 0.707i)10-s + (0.965 − 0.258i)12-s + 1.41i·13-s + (−0.999 − i)15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)18-s + (−1.22 − 0.707i)19-s + 1.41·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 + 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 + 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $0.925 + 0.378i$
Analytic conductor: \(0.586900\)
Root analytic conductor: \(0.766094\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (1157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1176,\ (\ :0),\ 0.925 + 0.378i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.951483533\)
\(L(\frac12)\) \(\approx\) \(1.951483533\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (-0.258 + 0.965i)T \)
7 \( 1 \)
good5 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 - 1.41iT - T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 - 2iT - T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + 1.41T + T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.582617778154516436555300638132, −8.746496036953196342868992308668, −8.384029524968738546175516709018, −7.18634097721022850889016654405, −6.55490742166348105346397605193, −5.78003267900940144079644900541, −4.86479144954023878925447198875, −4.01748821697190491750144985632, −2.50893210054384867237509584334, −1.65141091039412041623860924057, 2.15422116460045518234932830962, 2.97981725185121686199528370221, 3.69931932222236728200662875405, 4.79185693345783525580852991666, 5.78318015072267623302693791867, 6.25060424424820029161703157727, 7.42751763633024865822481047030, 8.528451343335756960993380730353, 9.681693025068087794429759884703, 10.28151766999378628448825098065

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