Properties

Label 2-912-57.53-c1-0-37
Degree $2$
Conductor $912$
Sign $-0.763 - 0.645i$
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  + (1.36 − 1.06i)3-s + (−2.20 − 2.62i)5-s + (−1.68 + 2.92i)7-s + (0.736 − 2.90i)9-s + (−2.33 + 1.34i)11-s + (−5.05 − 0.891i)13-s + (−5.81 − 1.24i)15-s + (−1.44 + 3.97i)17-s + (2.73 + 3.39i)19-s + (0.802 + 5.78i)21-s + (−1.69 + 2.01i)23-s + (−1.17 + 6.67i)25-s + (−2.08 − 4.75i)27-s + (3.54 − 1.28i)29-s + (−4.78 − 2.76i)31-s + ⋯
L(s)  = 1  + (0.789 − 0.614i)3-s + (−0.986 − 1.17i)5-s + (−0.637 + 1.10i)7-s + (0.245 − 0.969i)9-s + (−0.704 + 0.406i)11-s + (−1.40 − 0.247i)13-s + (−1.50 − 0.321i)15-s + (−0.350 + 0.963i)17-s + (0.626 + 0.779i)19-s + (0.175 + 1.26i)21-s + (−0.353 + 0.420i)23-s + (−0.235 + 1.33i)25-s + (−0.401 − 0.915i)27-s + (0.657 − 0.239i)29-s + (−0.858 − 0.495i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.763 - 0.645i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.763 - 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.763 - 0.645i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.763 - 0.645i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0122290 + 0.0334039i\)
\(L(\frac12)\) \(\approx\) \(0.0122290 + 0.0334039i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.36 + 1.06i)T \)
19 \( 1 + (-2.73 - 3.39i)T \)
good5 \( 1 + (2.20 + 2.62i)T + (-0.868 + 4.92i)T^{2} \)
7 \( 1 + (1.68 - 2.92i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.33 - 1.34i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (5.05 + 0.891i)T + (12.2 + 4.44i)T^{2} \)
17 \( 1 + (1.44 - 3.97i)T + (-13.0 - 10.9i)T^{2} \)
23 \( 1 + (1.69 - 2.01i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (-3.54 + 1.28i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (4.78 + 2.76i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 5.17iT - 37T^{2} \)
41 \( 1 + (-0.289 - 1.64i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (1.85 - 1.55i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (0.0440 + 0.120i)T + (-36.0 + 30.2i)T^{2} \)
53 \( 1 + (6.53 + 5.48i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (3.87 + 1.41i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-3.53 - 2.96i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (3.81 + 10.4i)T + (-51.3 + 43.0i)T^{2} \)
71 \( 1 + (9.91 - 8.31i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (0.414 + 2.35i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (2.22 - 0.391i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (6.27 + 3.62i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.209 + 1.18i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-3.13 + 8.60i)T + (-74.3 - 62.3i)T^{2} \)
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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

−9.392062944841059400638621588854, −8.654785608918228366164393907500, −7.901020890031317542606697215903, −7.42212477352728822695591298878, −6.08832945674363809029679455322, −5.14131682760296281077991326762, −4.06643149611571516928556098096, −2.96221916755637077582623724136, −1.86194471896492010898984290546, −0.01416216145880384140720470004, 2.70944068253108583121978456205, 3.15647551664756799365076429994, 4.21190510112418126145179859398, 5.03135745192029581650215464806, 6.79745889557529881517544672230, 7.28630957031056611378086923738, 7.85508119454741005350254866717, 9.029616620464629705195592694326, 9.947074709144916744347048358668, 10.46950507186732667730480651159

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