Properties

Label 2-912-76.3-c1-0-0
Degree $2$
Conductor $912$
Sign $-0.925 + 0.379i$
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 + (−0.467 + 2.64i)5-s + (−0.480 + 0.277i)7-s + (0.766 − 0.642i)9-s + (−1.68 − 0.971i)11-s + (−2.00 + 5.51i)13-s + (−0.467 − 2.64i)15-s + (−2.18 − 1.83i)17-s + (3.22 − 2.93i)19-s + (0.356 − 0.424i)21-s + (−3.17 + 0.559i)23-s + (−2.09 − 0.764i)25-s + (−0.500 + 0.866i)27-s + (−0.952 − 1.13i)29-s + (−3.09 − 5.35i)31-s + ⋯
L(s)  = 1  + (−0.542 + 0.197i)3-s + (−0.208 + 1.18i)5-s + (−0.181 + 0.104i)7-s + (0.255 − 0.214i)9-s + (−0.507 − 0.293i)11-s + (−0.556 + 1.52i)13-s + (−0.120 − 0.683i)15-s + (−0.530 − 0.445i)17-s + (0.739 − 0.672i)19-s + (0.0778 − 0.0927i)21-s + (−0.662 + 0.116i)23-s + (−0.419 − 0.152i)25-s + (−0.0962 + 0.166i)27-s + (−0.176 − 0.210i)29-s + (−0.555 − 0.962i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.925 + 0.379i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.925 + 0.379i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.925 + 0.379i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.925 + 0.379i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0521377 - 0.264350i\)
\(L(\frac12)\) \(\approx\) \(0.0521377 - 0.264350i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.22 + 2.93i)T \)
good5 \( 1 + (0.467 - 2.64i)T + (-4.69 - 1.71i)T^{2} \)
7 \( 1 + (0.480 - 0.277i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.68 + 0.971i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.00 - 5.51i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + (2.18 + 1.83i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (3.17 - 0.559i)T + (21.6 - 7.86i)T^{2} \)
29 \( 1 + (0.952 + 1.13i)T + (-5.03 + 28.5i)T^{2} \)
31 \( 1 + (3.09 + 5.35i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 6.18iT - 37T^{2} \)
41 \( 1 + (3.29 + 9.04i)T + (-31.4 + 26.3i)T^{2} \)
43 \( 1 + (2.72 + 0.481i)T + (40.4 + 14.7i)T^{2} \)
47 \( 1 + (2.27 + 2.71i)T + (-8.16 + 46.2i)T^{2} \)
53 \( 1 + (3.33 - 0.588i)T + (49.8 - 18.1i)T^{2} \)
59 \( 1 + (6.26 + 5.25i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (0.549 + 3.11i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-2.71 + 2.27i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-0.258 + 1.46i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (-5.13 + 1.87i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (10.7 - 3.92i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (12.9 - 7.47i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.23 - 11.6i)T + (-68.1 - 57.2i)T^{2} \)
97 \( 1 + (-4.52 + 5.39i)T + (-16.8 - 95.5i)T^{2} \)
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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

−10.65899621919948599988817006208, −9.767296348385652584736411383502, −9.148383483476248426571625708099, −7.83348464340937165038956375657, −6.96259914101548439631987816537, −6.48834258281622261725018765036, −5.34550901226868204364271235167, −4.35415029511022226380791706783, −3.24584724294939994083975842389, −2.15585419445229383750450239847, 0.13382744391788218085938731834, 1.54200808530356081303967327304, 3.15217555400450377440004557146, 4.47091264728993364679231310494, 5.24315532307859037917287677083, 5.91490485556232051328453137893, 7.19165496370161031791785051773, 7.968123983806979676708933110262, 8.634815166067848691903920992232, 9.805072334092559681885689033294

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