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Properties

Label 2-912-19.8-c2-0-5
Degree $2$
Conductor $912$
Sign $-0.247 - 0.968i$
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 + (4.62 − 8.01i)5-s + 1.36·7-s + (1.5 + 2.59i)9-s − 3.17·11-s + (−17.9 + 10.3i)13-s + (−13.8 + 8.01i)15-s + (−15.5 + 27.0i)17-s + (−14.8 + 11.8i)19-s + (−2.04 − 1.18i)21-s + (−0.507 − 0.878i)23-s + (−30.3 − 52.4i)25-s − 5.19i·27-s + (−3.56 + 2.05i)29-s + 41.8i·31-s + ⋯
L(s)  = 1  + (−0.5 − 0.288i)3-s + (0.925 − 1.60i)5-s + 0.194·7-s + (0.166 + 0.288i)9-s − 0.288·11-s + (−1.38 + 0.797i)13-s + (−0.925 + 0.534i)15-s + (−0.917 + 1.58i)17-s + (−0.780 + 0.625i)19-s + (−0.0973 − 0.0562i)21-s + (−0.0220 − 0.0382i)23-s + (−1.21 − 2.09i)25-s − 0.192i·27-s + (−0.122 + 0.0709i)29-s + 1.34i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.247 - 0.968i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.247 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.247 - 0.968i$
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: \((2,\ 912,\ (\ :1),\ -0.247 - 0.968i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.4068358641\)
\(L(\frac12)\) \(\approx\) \(0.4068358641\)
\(L(2)\) 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 \( 1 + (14.8 - 11.8i)T \)
good5 \( 1 + (-4.62 + 8.01i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 - 1.36T + 49T^{2} \)
11 \( 1 + 3.17T + 121T^{2} \)
13 \( 1 + (17.9 - 10.3i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + (15.5 - 27.0i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 + (0.507 + 0.878i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (3.56 - 2.05i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 - 41.8iT - 961T^{2} \)
37 \( 1 - 2.95iT - 1.36e3T^{2} \)
41 \( 1 + (-39.4 - 22.7i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-28.2 + 48.8i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-31.5 - 54.6i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-20.8 + 12.0i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (60.2 + 34.7i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (17.6 + 30.5i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (53.1 - 30.6i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (45.8 + 26.4i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-16.5 + 28.6i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (17.0 + 9.82i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 35.3T + 6.88e3T^{2} \)
89 \( 1 + (-18.8 + 10.8i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (26.1 + 15.1i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−10.12660316513275261506267022027, −9.204973255052964344661865058518, −8.606759477560785274042051627206, −7.70313070273156677871181753241, −6.48022375120468467312828076578, −5.79364712096244775348064587345, −4.82848827107660883315647635406, −4.30784242161712833844195561827, −2.17676295561479527942066427660, −1.47391731319901335738071129213, 0.13011461871277664679276983451, 2.37636447006661166785820352780, 2.80263036672585474752447118961, 4.40463265785409129383473139762, 5.37865814434443609653065528516, 6.18448065592440822188890114360, 7.09183396372749589270019796170, 7.58583846149871574165187463803, 9.201447137019297231167601857937, 9.774518146604734824478922887962

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