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Properties

Label 2-1056-11.3-c1-0-14
Degree $2$
Conductor $1056$
Sign $0.684 + 0.729i$
Analytic cond. $8.43220$
Root an. cond. $2.90382$
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  + (0.309 + 0.951i)3-s + (−1.02 − 0.741i)5-s + (0.167 − 0.516i)7-s + (−0.809 + 0.587i)9-s + (−0.493 − 3.27i)11-s + (2.16 − 1.57i)13-s + (0.389 − 1.19i)15-s + (3.76 + 2.73i)17-s + (0.283 + 0.872i)19-s + 0.543·21-s − 7.92·23-s + (−1.05 − 3.24i)25-s + (−0.809 − 0.587i)27-s + (3.02 − 9.32i)29-s + (2.67 − 1.94i)31-s + ⋯
L(s)  = 1  + (0.178 + 0.549i)3-s + (−0.456 − 0.331i)5-s + (0.0634 − 0.195i)7-s + (−0.269 + 0.195i)9-s + (−0.148 − 0.988i)11-s + (0.599 − 0.435i)13-s + (0.100 − 0.309i)15-s + (0.912 + 0.662i)17-s + (0.0650 + 0.200i)19-s + 0.118·21-s − 1.65·23-s + (−0.210 − 0.648i)25-s + (−0.155 − 0.113i)27-s + (0.562 − 1.73i)29-s + (0.479 − 0.348i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.684 + 0.729i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.684 + 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1056\)    =    \(2^{5} \cdot 3 \cdot 11\)
Sign: $0.684 + 0.729i$
Analytic conductor: \(8.43220\)
Root analytic conductor: \(2.90382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1056} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1056,\ (\ :1/2),\ 0.684 + 0.729i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.431817554\)
\(L(\frac12)\) \(\approx\) \(1.431817554\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.309 - 0.951i)T \)
11 \( 1 + (0.493 + 3.27i)T \)
good5 \( 1 + (1.02 + 0.741i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-0.167 + 0.516i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-2.16 + 1.57i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-3.76 - 2.73i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.283 - 0.872i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 7.92T + 23T^{2} \)
29 \( 1 + (-3.02 + 9.32i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.67 + 1.94i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-2.34 + 7.20i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.361 + 1.11i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 10.2T + 43T^{2} \)
47 \( 1 + (1.28 + 3.96i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-8.03 + 5.83i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.62 - 8.08i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (0.129 + 0.0942i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 2.62T + 67T^{2} \)
71 \( 1 + (-2.17 - 1.58i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (1.89 - 5.83i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (3.36 - 2.44i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (10.8 + 7.87i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 2.70T + 89T^{2} \)
97 \( 1 + (-6.61 + 4.80i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.01245796308229117043438680283, −8.827024566934084799361585747306, −8.145470585553441636239457976710, −7.68754319315687489079314182407, −6.03068285620330867067189985293, −5.71993431539801200159709388877, −4.18955073481724158968354888608, −3.81594298594705212093850614592, −2.49605044543786600708183199837, −0.69063931556861302462147975174, 1.38014687859750263621250813693, 2.63761840166575995819161505483, 3.68514967957689326681661136344, 4.78016657408845208545812531047, 5.85408797254183668003502866244, 6.83514083235758784882994446351, 7.48898840091773043149228175538, 8.215420073731057152582375169599, 9.158745355708715942615340873431, 9.977643148045338799649005278075

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