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Properties

Label 2-912-304.107-c1-0-47
Degree $2$
Conductor $912$
Sign $0.957 + 0.289i$
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  + (−0.841 + 1.13i)2-s + (0.965 + 0.258i)3-s + (−0.583 − 1.91i)4-s + (−0.192 + 0.719i)5-s + (−1.10 + 0.879i)6-s − 3.56·7-s + (2.66 + 0.947i)8-s + (0.866 + 0.499i)9-s + (−0.655 − 0.824i)10-s + (2.95 − 2.95i)11-s + (−0.0683 − 1.99i)12-s + (0.946 − 0.253i)13-s + (3.00 − 4.05i)14-s + (−0.372 + 0.644i)15-s + (−3.31 + 2.23i)16-s + (−2.56 − 4.44i)17-s + ⋯
L(s)  = 1  + (−0.595 + 0.803i)2-s + (0.557 + 0.149i)3-s + (−0.291 − 0.956i)4-s + (−0.0861 + 0.321i)5-s + (−0.451 + 0.359i)6-s − 1.34·7-s + (0.942 + 0.334i)8-s + (0.288 + 0.166i)9-s + (−0.207 − 0.260i)10-s + (0.892 − 0.892i)11-s + (−0.0197 − 0.577i)12-s + (0.262 − 0.0703i)13-s + (0.801 − 1.08i)14-s + (−0.0961 + 0.166i)15-s + (−0.829 + 0.558i)16-s + (−0.622 − 1.07i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.289i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.957 + 0.289i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (715, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.957 + 0.289i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01338 - 0.149839i\)
\(L(\frac12)\) \(\approx\) \(1.01338 - 0.149839i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.841 - 1.13i)T \)
3 \( 1 + (-0.965 - 0.258i)T \)
19 \( 1 + (1.65 + 4.03i)T \)
good5 \( 1 + (0.192 - 0.719i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + 3.56T + 7T^{2} \)
11 \( 1 + (-2.95 + 2.95i)T - 11iT^{2} \)
13 \( 1 + (-0.946 + 0.253i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (2.56 + 4.44i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-3.06 + 5.30i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (8.89 - 2.38i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 - 1.70T + 31T^{2} \)
37 \( 1 + (-3.96 + 3.96i)T - 37iT^{2} \)
41 \( 1 + (-1.87 - 3.24i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-9.50 - 2.54i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-4.03 - 2.33i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.10 + 7.86i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-3.51 + 13.1i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (0.430 + 1.60i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (2.22 + 8.30i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (1.97 - 1.14i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.81 + 1.62i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.68 - 2.91i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.09 - 3.09i)T + 83iT^{2} \)
89 \( 1 + (-5.67 + 9.82i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.01 - 4.62i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.520169609683347163722877054781, −9.258894363167952471489841968792, −8.591723620516167213816132065455, −7.38165617472697681751852208521, −6.70699343375860271355837353180, −6.12634676862615716095188848285, −4.83047152956539974004412403743, −3.64482657313544382547155307228, −2.58891104414562215640841289863, −0.60493212279680191552177727897, 1.34490384317238046821018102248, 2.51091569332640450614810908808, 3.75137717980546885276749741486, 4.18318016419611339916055975667, 5.99200318128466511765879060911, 6.96593474644437465975201649987, 7.72926360989178197930872899937, 8.889025742651949887458751468538, 9.199967889568297696728400251301, 10.00042696708901764691755179596

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