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Properties

Label 2-1332-37.2-c0-0-0
Degree $2$
Conductor $1332$
Sign $0.873 + 0.487i$
Analytic cond. $0.664754$
Root an. cond. $0.815324$
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  + (1.85 − 0.673i)7-s + (−0.939 − 1.34i)13-s + (−0.515 + 0.0451i)19-s + (0.642 + 0.766i)25-s + (−0.597 − 0.597i)31-s + (0.5 + 0.866i)37-s + (−1.40 + 1.40i)43-s + (2.20 − 1.85i)49-s + (1.15 − 0.811i)61-s + (0.524 + 1.43i)67-s − 1.53i·73-s + (0.0736 − 0.157i)79-s + (−2.64 − 1.85i)91-s + (0.515 + 1.92i)97-s + (−0.5 + 1.86i)103-s + ⋯
L(s)  = 1  + (1.85 − 0.673i)7-s + (−0.939 − 1.34i)13-s + (−0.515 + 0.0451i)19-s + (0.642 + 0.766i)25-s + (−0.597 − 0.597i)31-s + (0.5 + 0.866i)37-s + (−1.40 + 1.40i)43-s + (2.20 − 1.85i)49-s + (1.15 − 0.811i)61-s + (0.524 + 1.43i)67-s − 1.53i·73-s + (0.0736 − 0.157i)79-s + (−2.64 − 1.85i)91-s + (0.515 + 1.92i)97-s + (−0.5 + 1.86i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.873 + 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.873 + 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1332\)    =    \(2^{2} \cdot 3^{2} \cdot 37\)
Sign: $0.873 + 0.487i$
Analytic conductor: \(0.664754\)
Root analytic conductor: \(0.815324\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1332} (1297, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1332,\ (\ :0),\ 0.873 + 0.487i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.244487813\)
\(L(\frac12)\) \(\approx\) \(1.244487813\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
37 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (-0.642 - 0.766i)T^{2} \)
7 \( 1 + (-1.85 + 0.673i)T + (0.766 - 0.642i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.939 + 1.34i)T + (-0.342 + 0.939i)T^{2} \)
17 \( 1 + (0.342 + 0.939i)T^{2} \)
19 \( 1 + (0.515 - 0.0451i)T + (0.984 - 0.173i)T^{2} \)
23 \( 1 + (-0.866 + 0.5i)T^{2} \)
29 \( 1 + (-0.866 - 0.5i)T^{2} \)
31 \( 1 + (0.597 + 0.597i)T + iT^{2} \)
41 \( 1 + (0.939 + 0.342i)T^{2} \)
43 \( 1 + (1.40 - 1.40i)T - iT^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.766 + 0.642i)T^{2} \)
59 \( 1 + (0.642 - 0.766i)T^{2} \)
61 \( 1 + (-1.15 + 0.811i)T + (0.342 - 0.939i)T^{2} \)
67 \( 1 + (-0.524 - 1.43i)T + (-0.766 + 0.642i)T^{2} \)
71 \( 1 + (0.173 + 0.984i)T^{2} \)
73 \( 1 + 1.53iT - T^{2} \)
79 \( 1 + (-0.0736 + 0.157i)T + (-0.642 - 0.766i)T^{2} \)
83 \( 1 + (-0.939 + 0.342i)T^{2} \)
89 \( 1 + (-0.642 + 0.766i)T^{2} \)
97 \( 1 + (-0.515 - 1.92i)T + (-0.866 + 0.5i)T^{2} \)
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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

−9.926324758383409116539055704698, −8.795305771154110168948889508193, −7.86899562245977610530765631671, −7.69125641207217360742208235392, −6.55025601217139659162493751853, −5.20155837222902189536215190991, −4.91368697205723218659000058226, −3.77401048436069741270479138675, −2.47150732919114263874680433570, −1.24084188896274624280642742284, 1.72905186129401859337706077276, 2.43410113858641946287121224221, 4.11061941034522947012146270662, 4.83321656062107793627133931057, 5.50714778317462828764922993450, 6.73430463999861716760983550690, 7.46100969658141556416640213980, 8.470037252276384233910469305741, 8.822699257324894203508307389907, 9.847936957115270335439062319507

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