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Properties

Label 2-1176-7.6-c2-0-29
Degree $2$
Conductor $1176$
Sign $0.409 + 0.912i$
Analytic cond. $32.0436$
Root an. cond. $5.66071$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73i·3-s − 5.38i·5-s − 2.99·9-s + 20.7·11-s − 5.85i·13-s + 9.31·15-s − 29.1i·17-s + 29.3i·19-s − 15.0·23-s − 3.95·25-s − 5.19i·27-s − 40.8·29-s − 1.60i·31-s + 35.9i·33-s + 36.3·37-s + ⋯
L(s)  = 1  + 0.577i·3-s − 1.07i·5-s − 0.333·9-s + 1.88·11-s − 0.450i·13-s + 0.621·15-s − 1.71i·17-s + 1.54i·19-s − 0.654·23-s − 0.158·25-s − 0.192i·27-s − 1.40·29-s − 0.0517i·31-s + 1.08i·33-s + 0.982·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.409 + 0.912i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.409 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $0.409 + 0.912i$
Analytic conductor: \(32.0436\)
Root analytic conductor: \(5.66071\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1176,\ (\ :1),\ 0.409 + 0.912i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.873230004\)
\(L(\frac12)\) \(\approx\) \(1.873230004\)
\(L(2)\) 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 \( 1 - 1.73iT \)
7 \( 1 \)
good5 \( 1 + 5.38iT - 25T^{2} \)
11 \( 1 - 20.7T + 121T^{2} \)
13 \( 1 + 5.85iT - 169T^{2} \)
17 \( 1 + 29.1iT - 289T^{2} \)
19 \( 1 - 29.3iT - 361T^{2} \)
23 \( 1 + 15.0T + 529T^{2} \)
29 \( 1 + 40.8T + 841T^{2} \)
31 \( 1 + 1.60iT - 961T^{2} \)
37 \( 1 - 36.3T + 1.36e3T^{2} \)
41 \( 1 + 29.4iT - 1.68e3T^{2} \)
43 \( 1 + 26.6T + 1.84e3T^{2} \)
47 \( 1 + 48.6iT - 2.20e3T^{2} \)
53 \( 1 - 19.4T + 2.80e3T^{2} \)
59 \( 1 + 33.0iT - 3.48e3T^{2} \)
61 \( 1 + 81.3iT - 3.72e3T^{2} \)
67 \( 1 - 47.2T + 4.48e3T^{2} \)
71 \( 1 - 107.T + 5.04e3T^{2} \)
73 \( 1 - 40.9iT - 5.32e3T^{2} \)
79 \( 1 + 59.6T + 6.24e3T^{2} \)
83 \( 1 + 89.6iT - 6.88e3T^{2} \)
89 \( 1 + 11.1iT - 7.92e3T^{2} \)
97 \( 1 + 86.7iT - 9.40e3T^{2} \)
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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

−9.495581902182668319660413624090, −8.779695865928143901101291688706, −7.979979867237466141214197433804, −6.92256963539397359422301572500, −5.86785609491896850322460303320, −5.12500796902918241178954710103, −4.15717160983312778636632671111, −3.48506189753304900337216315960, −1.82106639744901101015787966774, −0.60899867469222505110693578388, 1.28398014918389147375986340318, 2.34310715359348329353132791870, 3.55455641661343617854376014637, 4.31408868220745940124836441525, 5.88660842820741071613896833946, 6.55604203314708824682346159631, 6.99375635135383882890731565510, 8.015050951915880000608363443114, 8.970615992142002417352126178111, 9.604472097707642700488412312258

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