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Properties

Label 2-912-76.71-c1-0-3
Degree $2$
Conductor $912$
Sign $0.861 - 0.508i$
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  + (0.173 − 0.984i)3-s + (−1.40 − 1.17i)5-s + (−4.46 + 2.57i)7-s + (−0.939 − 0.342i)9-s + (3.85 + 2.22i)11-s + (2.69 − 0.474i)13-s + (−1.40 + 1.17i)15-s + (−2.85 + 1.03i)17-s + (3.15 − 3.00i)19-s + (1.76 + 4.84i)21-s + (5.92 + 7.05i)23-s + (−0.283 − 1.60i)25-s + (−0.5 + 0.866i)27-s + (0.331 − 0.910i)29-s + (3.33 + 5.78i)31-s + ⋯
L(s)  = 1  + (0.100 − 0.568i)3-s + (−0.628 − 0.527i)5-s + (−1.68 + 0.974i)7-s + (−0.313 − 0.114i)9-s + (1.16 + 0.670i)11-s + (0.746 − 0.131i)13-s + (−0.362 + 0.304i)15-s + (−0.692 + 0.252i)17-s + (0.723 − 0.690i)19-s + (0.384 + 1.05i)21-s + (1.23 + 1.47i)23-s + (−0.0566 − 0.321i)25-s + (−0.0962 + 0.166i)27-s + (0.0615 − 0.169i)29-s + (0.599 + 1.03i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.861 - 0.508i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.861 - 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.861 - 0.508i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.861 - 0.508i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08735 + 0.296968i\)
\(L(\frac12)\) \(\approx\) \(1.08735 + 0.296968i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.173 + 0.984i)T \)
19 \( 1 + (-3.15 + 3.00i)T \)
good5 \( 1 + (1.40 + 1.17i)T + (0.868 + 4.92i)T^{2} \)
7 \( 1 + (4.46 - 2.57i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3.85 - 2.22i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.69 + 0.474i)T + (12.2 - 4.44i)T^{2} \)
17 \( 1 + (2.85 - 1.03i)T + (13.0 - 10.9i)T^{2} \)
23 \( 1 + (-5.92 - 7.05i)T + (-3.99 + 22.6i)T^{2} \)
29 \( 1 + (-0.331 + 0.910i)T + (-22.2 - 18.6i)T^{2} \)
31 \( 1 + (-3.33 - 5.78i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 6.33iT - 37T^{2} \)
41 \( 1 + (-6.98 - 1.23i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (1.22 - 1.45i)T + (-7.46 - 42.3i)T^{2} \)
47 \( 1 + (2.27 - 6.26i)T + (-36.0 - 30.2i)T^{2} \)
53 \( 1 + (-1.82 - 2.16i)T + (-9.20 + 52.1i)T^{2} \)
59 \( 1 + (9.47 - 3.44i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-0.706 + 0.592i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-14.4 - 5.25i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (9.76 + 8.19i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (-1.24 + 7.03i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (-0.463 + 2.63i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (7.41 - 4.28i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.927 - 0.163i)T + (83.6 - 30.4i)T^{2} \)
97 \( 1 + (-1.69 - 4.64i)T + (-74.3 + 62.3i)T^{2} \)
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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

−9.879031766147016819280521615077, −9.099781597883192492340049851996, −8.788137723246558160311839072380, −7.54185055557125726983579888401, −6.63097326258767428314432045735, −6.13696500037670475118602079617, −4.87345767733498060100869535987, −3.63965844965684852332564443057, −2.81092935839650923656693952724, −1.17575313385846522664611655774, 0.63517224865555128832276886305, 2.96721247258560931884979337911, 3.68425829543831595737883188812, 4.24026117049537537133877548713, 5.90373587732137488232496485634, 6.66294781990159748655470070683, 7.24976076619931534719014465886, 8.534014754580031111695461435996, 9.293391651253564348903068636249, 9.972740220310549448220993473826

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