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Properties

Label 2-912-19.17-c1-0-18
Degree $2$
Conductor $912$
Sign $-0.980 + 0.196i$
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 + (−1.22 + 0.444i)5-s + (−0.861 − 1.49i)7-s + (0.173 + 0.984i)9-s + (2.75 − 4.76i)11-s + (0.717 − 0.601i)13-s + (1.22 + 0.444i)15-s + (−0.682 + 3.86i)17-s + (0.168 + 4.35i)19-s + (−0.299 + 1.69i)21-s + (−7.45 − 2.71i)23-s + (−2.53 + 2.12i)25-s + (0.500 − 0.866i)27-s + (−0.194 − 1.10i)29-s + (−3.11 − 5.39i)31-s + ⋯
L(s)  = 1  + (−0.442 − 0.371i)3-s + (−0.546 + 0.198i)5-s + (−0.325 − 0.563i)7-s + (0.0578 + 0.328i)9-s + (0.830 − 1.43i)11-s + (0.198 − 0.166i)13-s + (0.315 + 0.114i)15-s + (−0.165 + 0.938i)17-s + (0.0387 + 0.999i)19-s + (−0.0652 + 0.370i)21-s + (−1.55 − 0.566i)23-s + (−0.506 + 0.425i)25-s + (0.0962 − 0.166i)27-s + (−0.0361 − 0.204i)29-s + (−0.559 − 0.968i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 + 0.196i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.980 + 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.980 + 0.196i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (625, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.980 + 0.196i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0410816 - 0.414734i\)
\(L(\frac12)\) \(\approx\) \(0.0410816 - 0.414734i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.168 - 4.35i)T \)
good5 \( 1 + (1.22 - 0.444i)T + (3.83 - 3.21i)T^{2} \)
7 \( 1 + (0.861 + 1.49i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.75 + 4.76i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.717 + 0.601i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (0.682 - 3.86i)T + (-15.9 - 5.81i)T^{2} \)
23 \( 1 + (7.45 + 2.71i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (0.194 + 1.10i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (3.11 + 5.39i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 1.20T + 37T^{2} \)
41 \( 1 + (9.01 + 7.56i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (8.88 - 3.23i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (1.53 + 8.73i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + (3.90 + 1.42i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (-0.174 + 0.991i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-2.88 - 1.05i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-2.23 - 12.6i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (12.3 - 4.49i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (-3.29 - 2.76i)T + (12.6 + 71.8i)T^{2} \)
79 \( 1 + (-0.476 - 0.399i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-1.52 - 2.63i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.29 + 1.08i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (-0.416 + 2.35i)T + (-91.1 - 33.1i)T^{2} \)
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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

−9.964864883716998033776095241020, −8.559382001617911845779951435159, −8.120416505114967731479598003492, −7.05248067950126292379876667302, −6.22001191163182179410809336590, −5.61269334349235553491706890950, −3.92368262329471813044884911926, −3.61004413750097760812718651266, −1.74363087963355353728699829697, −0.20580925019642983326386000798, 1.81351752937256926367954146808, 3.30863748049357646287818964490, 4.40533222583321352929021674462, 5.03052247562340862600162175552, 6.27551294275946515268560907053, 6.97231809455286861178059302901, 7.937617650866468769299621770957, 9.086335181027975552965851695532, 9.549853599512586097481431696242, 10.38410271232223379852641434761

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