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Properties

Label 2-912-57.56-c1-0-35
Degree $2$
Conductor $912$
Sign $-0.993 - 0.114i$
Analytic cond. $7.28235$
Root an. cond. $2.69858$
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  + (−1.5 − 0.866i)3-s − 3.46i·5-s − 7-s + (1.5 + 2.59i)9-s − 3.46i·11-s − 1.73i·13-s + (−2.99 + 5.19i)15-s − 1.73i·17-s + (4 − 1.73i)19-s + (1.5 + 0.866i)21-s − 5.19i·23-s − 6.99·25-s − 5.19i·27-s − 9·29-s + 10.3i·31-s + ⋯
L(s)  = 1  + (−0.866 − 0.499i)3-s − 1.54i·5-s − 0.377·7-s + (0.5 + 0.866i)9-s − 1.04i·11-s − 0.480i·13-s + (−0.774 + 1.34i)15-s − 0.420i·17-s + (0.917 − 0.397i)19-s + (0.327 + 0.188i)21-s − 1.08i·23-s − 1.39·25-s − 0.999i·27-s − 1.67·29-s + 1.86i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.114i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 - 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.993 - 0.114i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.993 - 0.114i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0408400 + 0.709720i\)
\(L(\frac12)\) \(\approx\) \(0.0408400 + 0.709720i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.5 + 0.866i)T \)
19 \( 1 + (-4 + 1.73i)T \)
good5 \( 1 + 3.46iT - 5T^{2} \)
7 \( 1 + T + 7T^{2} \)
11 \( 1 + 3.46iT - 11T^{2} \)
13 \( 1 + 1.73iT - 13T^{2} \)
17 \( 1 + 1.73iT - 17T^{2} \)
23 \( 1 + 5.19iT - 23T^{2} \)
29 \( 1 + 9T + 29T^{2} \)
31 \( 1 - 10.3iT - 31T^{2} \)
37 \( 1 - 6.92iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 2T + 43T^{2} \)
47 \( 1 - 3.46iT - 47T^{2} \)
53 \( 1 - 9T + 53T^{2} \)
59 \( 1 + 3T + 59T^{2} \)
61 \( 1 + 8T + 61T^{2} \)
67 \( 1 + 8.66iT - 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 - 11T + 73T^{2} \)
79 \( 1 + 6.92iT - 79T^{2} \)
83 \( 1 - 10.3iT - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + 13.8iT - 97T^{2} \)
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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

−9.594327308765145281382590752132, −8.758646213835674476160718874746, −8.050569957155588787338996736996, −7.04780362551873855628173892357, −6.01787168329570719286958867729, −5.27897725139969926680995703297, −4.64152189035303064600392259549, −3.17224876554612289124470968074, −1.41257393692893365059355368389, −0.39420738451292526617359265397, 2.00054907663525184032548186890, 3.43549569003335863127305592153, 4.12571298516247816773420089958, 5.51609030431458289553462675499, 6.16856297446901935037821256248, 7.16953367436832341250413445637, 7.51993362264571322451593200575, 9.358588007630277367990214762527, 9.778709980490372354394574336033, 10.50687907800007814205506569116

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