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Properties

Label 2-912-76.51-c1-0-6
Degree $2$
Conductor $912$
Sign $0.505 - 0.862i$
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  + (−0.939 − 0.342i)3-s + (0.494 + 2.80i)5-s + (4.05 + 2.33i)7-s + (0.766 + 0.642i)9-s + (4.23 − 2.44i)11-s + (−0.974 − 2.67i)13-s + (0.494 − 2.80i)15-s + (−3.09 + 2.59i)17-s + (−3.35 + 2.78i)19-s + (−3.00 − 3.58i)21-s + (5.98 + 1.05i)23-s + (−2.92 + 1.06i)25-s + (−0.500 − 0.866i)27-s + (−3.64 + 4.34i)29-s + (3.14 − 5.44i)31-s + ⋯
L(s)  = 1  + (−0.542 − 0.197i)3-s + (0.221 + 1.25i)5-s + (1.53 + 0.884i)7-s + (0.255 + 0.214i)9-s + (1.27 − 0.737i)11-s + (−0.270 − 0.742i)13-s + (0.127 − 0.724i)15-s + (−0.751 + 0.630i)17-s + (−0.770 + 0.637i)19-s + (−0.656 − 0.782i)21-s + (1.24 + 0.220i)23-s + (−0.585 + 0.213i)25-s + (−0.0962 − 0.166i)27-s + (−0.676 + 0.806i)29-s + (0.564 − 0.977i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.505 - 0.862i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.505 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.505 - 0.862i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.505 - 0.862i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.44684 + 0.828705i\)
\(L(\frac12)\) \(\approx\) \(1.44684 + 0.828705i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.939 + 0.342i)T \)
19 \( 1 + (3.35 - 2.78i)T \)
good5 \( 1 + (-0.494 - 2.80i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (-4.05 - 2.33i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-4.23 + 2.44i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.974 + 2.67i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (3.09 - 2.59i)T + (2.95 - 16.7i)T^{2} \)
23 \( 1 + (-5.98 - 1.05i)T + (21.6 + 7.86i)T^{2} \)
29 \( 1 + (3.64 - 4.34i)T + (-5.03 - 28.5i)T^{2} \)
31 \( 1 + (-3.14 + 5.44i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 6.42iT - 37T^{2} \)
41 \( 1 + (2.01 - 5.54i)T + (-31.4 - 26.3i)T^{2} \)
43 \( 1 + (-1.67 + 0.296i)T + (40.4 - 14.7i)T^{2} \)
47 \( 1 + (5.46 - 6.50i)T + (-8.16 - 46.2i)T^{2} \)
53 \( 1 + (-7.07 - 1.24i)T + (49.8 + 18.1i)T^{2} \)
59 \( 1 + (-3.61 + 3.03i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-0.715 + 4.05i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (2.79 + 2.34i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-0.152 - 0.864i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (9.38 + 3.41i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (-12.2 - 4.45i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (4.28 + 2.47i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-3.03 - 8.34i)T + (-68.1 + 57.2i)T^{2} \)
97 \( 1 + (-11.7 - 14.0i)T + (-16.8 + 95.5i)T^{2} \)
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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

−10.65720375316919373082647821087, −9.342594229530736821661171448998, −8.497990440897260021323371079367, −7.68840783490237374198435210208, −6.63476473876302079764160085862, −6.01170946494795756689334621712, −5.13650120581243596742936323084, −3.93925560927125720809935724866, −2.62058131222653817407175184666, −1.52715520199645121885409312097, 0.980108433772262211726696456745, 1.90174637440377008758728972504, 4.19816490526195898103613197404, 4.60223520887331508037482864675, 5.19085562869104778707803795382, 6.71010753510946963918678807895, 7.21208238716042103970347787869, 8.576969994248576022287860223863, 8.969801085565638612723550814129, 9.928057632507909563090734907294

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