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Properties

Label 2-912-19.8-c2-0-34
Degree $2$
Conductor $912$
Sign $0.975 + 0.217i$
Analytic cond. $24.8502$
Root an. cond. $4.98499$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.5 + 0.866i)3-s + (2.32 − 4.03i)5-s + 10.6·7-s + (1.5 + 2.59i)9-s + 6.37·11-s + (15.3 − 8.87i)13-s + (6.98 − 4.03i)15-s + (−5.84 + 10.1i)17-s + (3.88 + 18.5i)19-s + (16.0 + 9.25i)21-s + (−13.9 − 24.2i)23-s + (1.66 + 2.88i)25-s + 5.19i·27-s + (−33.2 + 19.1i)29-s − 42.9i·31-s + ⋯
L(s)  = 1  + (0.5 + 0.288i)3-s + (0.465 − 0.806i)5-s + 1.52·7-s + (0.166 + 0.288i)9-s + 0.579·11-s + (1.18 − 0.682i)13-s + (0.465 − 0.268i)15-s + (−0.344 + 0.596i)17-s + (0.204 + 0.978i)19-s + (0.762 + 0.440i)21-s + (−0.607 − 1.05i)23-s + (0.0667 + 0.115i)25-s + 0.192i·27-s + (−1.14 + 0.661i)29-s − 1.38i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.217i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.975 + 0.217i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.975 + 0.217i$
Analytic conductor: \(24.8502\)
Root analytic conductor: \(4.98499\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (673, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1),\ 0.975 + 0.217i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.214895841\)
\(L(\frac12)\) \(\approx\) \(3.214895841\)
\(L(2)\) 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 \( 1 + (-1.5 - 0.866i)T \)
19 \( 1 + (-3.88 - 18.5i)T \)
good5 \( 1 + (-2.32 + 4.03i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 - 10.6T + 49T^{2} \)
11 \( 1 - 6.37T + 121T^{2} \)
13 \( 1 + (-15.3 + 8.87i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + (5.84 - 10.1i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 + (13.9 + 24.2i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (33.2 - 19.1i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + 42.9iT - 961T^{2} \)
37 \( 1 + 33.9iT - 1.36e3T^{2} \)
41 \( 1 + (-16.3 - 9.45i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (26.5 - 45.9i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-12.0 - 20.8i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (13.3 - 7.69i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (25.6 + 14.8i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (21.3 + 36.9i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-15.1 + 8.75i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (-74.6 - 43.0i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-46.2 + 80.0i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-26.3 - 15.2i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 77.1T + 6.88e3T^{2} \)
89 \( 1 + (76.8 - 44.3i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (1.82 + 1.05i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−9.706638901368649406751836068242, −8.926187421826970573749927096448, −8.229510939450865087918690070931, −7.76117935448141307069602341712, −6.19910551526800305429767291420, −5.44206830131064823019774182412, −4.45977674064496134637637520605, −3.68592835071177982053350655042, −2.00586518643531680793137913780, −1.22801405318415385105632395657, 1.37250589518619758881907597632, 2.19640976377351877891036936244, 3.47706200104227061997900524911, 4.51890201106753202623596182500, 5.62754522371517335369872540488, 6.69582824053514936411922219291, 7.29702464236888574259412777368, 8.347069893605135253624248360073, 8.949637687631210597391196310212, 9.847889495925002963158302795742

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