Properties

Label 2-912-57.56-c1-0-9
Degree $2$
Conductor $912$
Sign $-0.0215 - 0.999i$
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  + (1.39 + 1.02i)3-s − 1.12i·5-s − 3.21·7-s + (0.886 + 2.86i)9-s + 3.06i·11-s + 0.110i·13-s + (1.15 − 1.56i)15-s + 1.89i·17-s + (2.66 + 3.45i)19-s + (−4.48 − 3.30i)21-s + 7.89i·23-s + 3.73·25-s + (−1.71 + 4.90i)27-s − 1.77·29-s + 5.07i·31-s + ⋯
L(s)  = 1  + (0.804 + 0.593i)3-s − 0.502i·5-s − 1.21·7-s + (0.295 + 0.955i)9-s + 0.925i·11-s + 0.0306i·13-s + (0.298 − 0.404i)15-s + 0.460i·17-s + (0.610 + 0.791i)19-s + (−0.978 − 0.721i)21-s + 1.64i·23-s + 0.747·25-s + (−0.329 + 0.944i)27-s − 0.329·29-s + 0.911i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0215 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0215 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.0215 - 0.999i$
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: \((2,\ 912,\ (\ :1/2),\ -0.0215 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13796 + 1.16271i\)
\(L(\frac12)\) \(\approx\) \(1.13796 + 1.16271i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.39 - 1.02i)T \)
19 \( 1 + (-2.66 - 3.45i)T \)
good5 \( 1 + 1.12iT - 5T^{2} \)
7 \( 1 + 3.21T + 7T^{2} \)
11 \( 1 - 3.06iT - 11T^{2} \)
13 \( 1 - 0.110iT - 13T^{2} \)
17 \( 1 - 1.89iT - 17T^{2} \)
23 \( 1 - 7.89iT - 23T^{2} \)
29 \( 1 + 1.77T + 29T^{2} \)
31 \( 1 - 5.07iT - 31T^{2} \)
37 \( 1 + 3.59iT - 37T^{2} \)
41 \( 1 - 1.01T + 41T^{2} \)
43 \( 1 - 8.31T + 43T^{2} \)
47 \( 1 + 5.68iT - 47T^{2} \)
53 \( 1 - 0.535T + 53T^{2} \)
59 \( 1 + 12.5T + 59T^{2} \)
61 \( 1 - 7.02T + 61T^{2} \)
67 \( 1 + 12.5iT - 67T^{2} \)
71 \( 1 + 2.76T + 71T^{2} \)
73 \( 1 + 5.47T + 73T^{2} \)
79 \( 1 + 17.5iT - 79T^{2} \)
83 \( 1 - 8.67iT - 83T^{2} \)
89 \( 1 + 2.12T + 89T^{2} \)
97 \( 1 + 12.8iT - 97T^{2} \)
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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

−10.04213143878808192639332091936, −9.441884228697117295494103178542, −8.913850939043940013566395626881, −7.78530305470037332468347363836, −7.11772597793490803856290935951, −5.87026137197600497917106887473, −4.91286959096148731017847587880, −3.84308053570042253467160338746, −3.11693673828595407411233480764, −1.72140823064859246616010088569, 0.70582228707070036097261147945, 2.68879618116951309500498719687, 3.04301228004406467764099310199, 4.25741969924202448060628084990, 5.84393406455390507556499136682, 6.61474488600082381106067311885, 7.21206574959027565891066527616, 8.241726235142930615235055741880, 9.070480906513144426354335507211, 9.673987301315312470855562764609

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