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Properties

Label 2-912-76.27-c1-0-18
Degree $2$
Conductor $912$
Sign $-0.977 + 0.211i$
Analytic cond. $7.28235$
Root an. cond. $2.69858$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)3-s − 1.73i·7-s + (−0.499 − 0.866i)9-s − 3.46i·11-s + (−4.5 + 2.59i)13-s + (−3 + 5.19i)17-s + (−4 − 1.73i)19-s + (1.49 + 0.866i)21-s + (−3 + 1.73i)23-s + (2.5 + 4.33i)25-s + 0.999·27-s + (−3 + 1.73i)29-s + 31-s + (2.99 + 1.73i)33-s − 8.66i·37-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s − 0.654i·7-s + (−0.166 − 0.288i)9-s − 1.04i·11-s + (−1.24 + 0.720i)13-s + (−0.727 + 1.26i)17-s + (−0.917 − 0.397i)19-s + (0.327 + 0.188i)21-s + (−0.625 + 0.361i)23-s + (0.5 + 0.866i)25-s + 0.192·27-s + (−0.557 + 0.321i)29-s + 0.179·31-s + (0.522 + 0.301i)33-s − 1.42i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 + 0.211i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.977 + 0.211i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.977 + 0.211i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (4 + 1.73i)T \)
good5 \( 1 + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 1.73iT - 7T^{2} \)
11 \( 1 + 3.46iT - 11T^{2} \)
13 \( 1 + (4.5 - 2.59i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (3 - 5.19i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (3 - 1.73i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3 - 1.73i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - T + 31T^{2} \)
37 \( 1 + 8.66iT - 37T^{2} \)
41 \( 1 + (6 + 3.46i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.5 + 2.59i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (9 - 5.19i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (9 - 5.19i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6 + 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.5 - 4.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.5 + 11.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.5 + 4.33i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 17.3iT - 83T^{2} \)
89 \( 1 + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-12 - 6.92i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.704001764668296311560330772903, −8.977373359005679595698539748583, −8.098259485724013227670015459427, −7.02627019948769278312967309926, −6.27856350878453644979430624113, −5.22027614722570624055447747227, −4.27982175837831834577418667677, −3.46189218686057917093529748649, −1.93872749864260990693501696061, 0, 2.00648246020510612466031685794, 2.80931720587868775372248686869, 4.56172340174618364993124098657, 5.12919997442564396040660016979, 6.33783642338348366719625177894, 7.00717738142742386289386435993, 7.932125409435199671065660267730, 8.700241594274867753620935221654, 9.872141008160081618662136574981

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