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Properties

Label 2-912-76.67-c1-0-14
Degree $2$
Conductor $912$
Sign $0.999 + 0.0193i$
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  + (0.766 − 0.642i)3-s + (3.79 + 1.38i)5-s + (0.699 + 0.404i)7-s + (0.173 − 0.984i)9-s + (5.19 − 3.00i)11-s + (−3.41 + 4.06i)13-s + (3.79 − 1.38i)15-s + (0.0141 + 0.0800i)17-s + (−2.65 − 3.45i)19-s + (0.795 − 0.140i)21-s + (0.391 + 1.07i)23-s + (8.68 + 7.28i)25-s + (−0.500 − 0.866i)27-s + (−9.02 − 1.59i)29-s + (−0.580 + 1.00i)31-s + ⋯
L(s)  = 1  + (0.442 − 0.371i)3-s + (1.69 + 0.618i)5-s + (0.264 + 0.152i)7-s + (0.0578 − 0.328i)9-s + (1.56 − 0.905i)11-s + (−0.946 + 1.12i)13-s + (0.980 − 0.356i)15-s + (0.00342 + 0.0194i)17-s + (−0.609 − 0.792i)19-s + (0.173 − 0.0306i)21-s + (0.0816 + 0.224i)23-s + (1.73 + 1.45i)25-s + (−0.0962 − 0.166i)27-s + (−1.67 − 0.295i)29-s + (−0.104 + 0.180i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0193i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0193i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.999 + 0.0193i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (751, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.999 + 0.0193i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.56912 - 0.0248785i\)
\(L(\frac12)\) \(\approx\) \(2.56912 - 0.0248785i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.766 + 0.642i)T \)
19 \( 1 + (2.65 + 3.45i)T \)
good5 \( 1 + (-3.79 - 1.38i)T + (3.83 + 3.21i)T^{2} \)
7 \( 1 + (-0.699 - 0.404i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-5.19 + 3.00i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.41 - 4.06i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + (-0.0141 - 0.0800i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (-0.391 - 1.07i)T + (-17.6 + 14.7i)T^{2} \)
29 \( 1 + (9.02 + 1.59i)T + (27.2 + 9.91i)T^{2} \)
31 \( 1 + (0.580 - 1.00i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 1.82iT - 37T^{2} \)
41 \( 1 + (-3.31 - 3.94i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (1.60 - 4.41i)T + (-32.9 - 27.6i)T^{2} \)
47 \( 1 + (8.24 + 1.45i)T + (44.1 + 16.0i)T^{2} \)
53 \( 1 + (-3.79 - 10.4i)T + (-40.6 + 34.0i)T^{2} \)
59 \( 1 + (1.58 + 8.99i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-3.36 + 1.22i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-1.28 + 7.30i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (4.80 + 1.74i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (-0.489 + 0.410i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (-4.13 + 3.46i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-9.34 - 5.39i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.59 + 5.47i)T + (-15.4 - 87.6i)T^{2} \)
97 \( 1 + (7.85 - 1.38i)T + (91.1 - 33.1i)T^{2} \)
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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

−9.720286948180705438492493943622, −9.354758799301636874595900788729, −8.717261936414599258756943072579, −7.34132416588425627553146976025, −6.52765513555794221173140975127, −6.08448638909204575529220843753, −4.87003079320314583893917980182, −3.53151286593747231784618731026, −2.31766635638563525839186222622, −1.59386900181270316358172926298, 1.50257892854022113715455094741, 2.32663474168924779907600906914, 3.83612564761411815625547688476, 4.87903497913421721301570122527, 5.62265960097653185933660113082, 6.58123216149337676402348200641, 7.60979760163466115854980531437, 8.717372230112496945103764262308, 9.372509816972369799840430246565, 9.934440907165808276640322536009

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