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Properties

Label 2-912-19.4-c1-0-4
Degree $2$
Conductor $912$
Sign $-0.962 - 0.269i$
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 + (−0.0923 + 0.0775i)5-s + (−2.14 + 3.71i)7-s + (−0.939 + 0.342i)9-s + (1.28 + 2.22i)11-s + (0.141 − 0.802i)13-s + (−0.0923 − 0.0775i)15-s + (0.439 + 0.160i)17-s + (−3.16 − 2.99i)19-s + (−4.03 − 1.46i)21-s + (−4.25 − 3.57i)23-s + (−0.865 + 4.90i)25-s + (−0.5 − 0.866i)27-s + (−2.20 + 0.802i)29-s + (2.67 − 4.63i)31-s + ⋯
L(s)  = 1  + (0.100 + 0.568i)3-s + (−0.0413 + 0.0346i)5-s + (−0.810 + 1.40i)7-s + (−0.313 + 0.114i)9-s + (0.388 + 0.672i)11-s + (0.0392 − 0.222i)13-s + (−0.0238 − 0.0200i)15-s + (0.106 + 0.0388i)17-s + (−0.726 − 0.687i)19-s + (−0.879 − 0.320i)21-s + (−0.887 − 0.744i)23-s + (−0.173 + 0.981i)25-s + (−0.0962 − 0.166i)27-s + (−0.409 + 0.149i)29-s + (0.480 − 0.831i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 - 0.269i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.962 - 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.962 - 0.269i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.962 - 0.269i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.115081 + 0.837602i\)
\(L(\frac12)\) \(\approx\) \(0.115081 + 0.837602i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.16 + 2.99i)T \)
good5 \( 1 + (0.0923 - 0.0775i)T + (0.868 - 4.92i)T^{2} \)
7 \( 1 + (2.14 - 3.71i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.28 - 2.22i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.141 + 0.802i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (-0.439 - 0.160i)T + (13.0 + 10.9i)T^{2} \)
23 \( 1 + (4.25 + 3.57i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (2.20 - 0.802i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-2.67 + 4.63i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 8.51T + 37T^{2} \)
41 \( 1 + (-0.666 - 3.77i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (7.14 - 5.99i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (8.90 - 3.24i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (-9.77 - 8.20i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (-14.1 - 5.14i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (1.31 + 1.10i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-10.7 + 3.91i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (10.2 - 8.57i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (-0.396 - 2.24i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (0.843 + 4.78i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-1.62 + 2.81i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.595 + 3.37i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (2.91 + 1.06i)T + (74.3 + 62.3i)T^{2} \)
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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

−10.22653015002000705314874136756, −9.633545884874226831154796828450, −8.911474534094592596032280695204, −8.220496377233648675085623156468, −6.92683071684370859214028740889, −6.11056636993585085067534005577, −5.26339731004637606498042926814, −4.20976033835653443962388958444, −3.09911557520587182029318220371, −2.12577680163078965239831157024, 0.37637696916551082606209091099, 1.82632904728196491194209253247, 3.46198727058996194710240467539, 3.97838737488322521595321320866, 5.46953627829039934465290992613, 6.57797425217815204258330911565, 6.93885512358643401920636284242, 8.058584148352216679484335815200, 8.687743373645110358193473445239, 9.981211851187371341570400037981

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