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Properties

Label 2-912-76.27-c1-0-11
Degree $2$
Conductor $912$
Sign $0.879 + 0.475i$
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  + (−0.5 + 0.866i)3-s + (−0.675 + 1.17i)5-s − 1.45i·7-s + (−0.499 − 0.866i)9-s − 3.18i·11-s + (−2.23 + 1.28i)13-s + (−0.675 − 1.17i)15-s + (2.08 − 3.61i)17-s + (2.43 − 3.61i)19-s + (1.26 + 0.728i)21-s + (6.49 − 3.75i)23-s + (1.58 + 2.74i)25-s + 0.999·27-s + (−0.734 + 0.423i)29-s − 0.351·31-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (−0.302 + 0.523i)5-s − 0.550i·7-s + (−0.166 − 0.288i)9-s − 0.961i·11-s + (−0.619 + 0.357i)13-s + (−0.174 − 0.302i)15-s + (0.505 − 0.876i)17-s + (0.559 − 0.828i)19-s + (0.275 + 0.159i)21-s + (1.35 − 0.782i)23-s + (0.317 + 0.549i)25-s + 0.192·27-s + (−0.136 + 0.0787i)29-s − 0.0632·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.879 + 0.475i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.879 + 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.879 + 0.475i$
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: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.879 + 0.475i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18015 - 0.298632i\)
\(L(\frac12)\) \(\approx\) \(1.18015 - 0.298632i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.43 + 3.61i)T \)
good5 \( 1 + (0.675 - 1.17i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 1.45iT - 7T^{2} \)
11 \( 1 + 3.18iT - 11T^{2} \)
13 \( 1 + (2.23 - 1.28i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.08 + 3.61i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-6.49 + 3.75i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.734 - 0.423i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.351T + 31T^{2} \)
37 \( 1 - 6.89iT - 37T^{2} \)
41 \( 1 + (4.05 + 2.34i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.52 + 3.76i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-8.04 + 4.64i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-8.76 + 5.05i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.675 + 1.17i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.29 - 7.43i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.08 - 1.88i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.438 + 0.758i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-6.67 + 11.5i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.52 + 4.37i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 7.22iT - 83T^{2} \)
89 \( 1 + (3.81 - 2.20i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.79 + 4.49i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.16596685776051519587379878442, −9.235994605931572884305387746242, −8.482967150247555028404804472057, −7.14428878908503747662407839119, −6.91303506064909299263224028937, −5.48245425682134070747435756692, −4.78636697960137146329752975074, −3.57607587867932585985733948714, −2.80040300193354369364332505204, −0.69779571208406653154871227506, 1.24055546689813708741794607375, 2.51939945334639758462380931545, 3.88905690458774376704939837867, 5.08306015659375717262049470041, 5.65968812777049633873332805522, 6.85966566735469927651047170203, 7.63270117770910667133264273143, 8.362962847454455034936147218195, 9.332676524673843855806132216032, 10.10632590427527828237385664237

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