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Properties

Label 2-1176-21.20-c1-0-7
Degree $2$
Conductor $1176$
Sign $0.901 - 0.433i$
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  + (0.454 − 1.67i)3-s − 2.80·5-s + (−2.58 − 1.52i)9-s + 5.48i·11-s + 1.35i·13-s + (−1.27 + 4.69i)15-s + 5.77·17-s − 1.98i·19-s + 2.42i·23-s + 2.88·25-s + (−3.71 + 3.63i)27-s + 7.05i·29-s − 3.55i·31-s + (9.15 + 2.49i)33-s + 4.28·37-s + ⋯
L(s)  = 1  + (0.262 − 0.964i)3-s − 1.25·5-s + (−0.862 − 0.506i)9-s + 1.65i·11-s + 0.376i·13-s + (−0.329 + 1.21i)15-s + 1.40·17-s − 0.455i·19-s + 0.505i·23-s + 0.576·25-s + (−0.715 + 0.698i)27-s + 1.31i·29-s − 0.637i·31-s + (1.59 + 0.434i)33-s + 0.704·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.901 - 0.433i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.901 - 0.433i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $0.901 - 0.433i$
Analytic conductor: \(9.39040\)
Root analytic conductor: \(3.06437\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1176,\ (\ :1/2),\ 0.901 - 0.433i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.155082751\)
\(L(\frac12)\) \(\approx\) \(1.155082751\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.454 + 1.67i)T \)
7 \( 1 \)
good5 \( 1 + 2.80T + 5T^{2} \)
11 \( 1 - 5.48iT - 11T^{2} \)
13 \( 1 - 1.35iT - 13T^{2} \)
17 \( 1 - 5.77T + 17T^{2} \)
19 \( 1 + 1.98iT - 19T^{2} \)
23 \( 1 - 2.42iT - 23T^{2} \)
29 \( 1 - 7.05iT - 29T^{2} \)
31 \( 1 + 3.55iT - 31T^{2} \)
37 \( 1 - 4.28T + 37T^{2} \)
41 \( 1 - 1.81T + 41T^{2} \)
43 \( 1 - 11.2T + 43T^{2} \)
47 \( 1 + 0.402T + 47T^{2} \)
53 \( 1 - 6.09iT - 53T^{2} \)
59 \( 1 - 2.56T + 59T^{2} \)
61 \( 1 - 5.49iT - 61T^{2} \)
67 \( 1 + 6.90T + 67T^{2} \)
71 \( 1 - 2.08iT - 71T^{2} \)
73 \( 1 + 0.341iT - 73T^{2} \)
79 \( 1 + 2.38T + 79T^{2} \)
83 \( 1 - 11.8T + 83T^{2} \)
89 \( 1 + 1.15T + 89T^{2} \)
97 \( 1 - 16.0iT - 97T^{2} \)
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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

−9.644561832193026450766668480087, −8.949127846029615311524582802209, −7.77914855593144155310402844836, −7.55084289878633718525070915837, −6.85543053610089965215161225441, −5.67767858927280932289514279024, −4.53572780070016449470826045851, −3.62079432442009936512741807432, −2.49116886299038131896331185495, −1.17531457944528000313668458781, 0.56908269731542886082684771157, 2.87159867756713171849853692909, 3.57051504033469642979871278610, 4.26960144132942507520728010142, 5.43868754423999000302365134155, 6.12590286575267747785819214300, 7.69159408159948121384217688905, 8.074574304626422820014688114661, 8.778546194306547991254877224678, 9.755987946824060514368568471060

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