Properties

Label 2-912-76.71-c1-0-2
Degree $2$
Conductor $912$
Sign $0.748 - 0.663i$
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 + (−3.37 − 2.82i)5-s + (−1.97 + 1.14i)7-s + (−0.939 − 0.342i)9-s + (0.618 + 0.357i)11-s + (−1.61 + 0.285i)13-s + (3.37 − 2.82i)15-s + (4.33 − 1.57i)17-s + (4.33 + 0.407i)19-s + (−0.779 − 2.14i)21-s + (4.29 + 5.12i)23-s + (2.49 + 14.1i)25-s + (0.5 − 0.866i)27-s + (0.0802 − 0.220i)29-s + (0.525 + 0.910i)31-s + ⋯
L(s)  = 1  + (−0.100 + 0.568i)3-s + (−1.50 − 1.26i)5-s + (−0.746 + 0.430i)7-s + (−0.313 − 0.114i)9-s + (0.186 + 0.107i)11-s + (−0.448 + 0.0790i)13-s + (0.870 − 0.730i)15-s + (1.05 − 0.382i)17-s + (0.995 + 0.0934i)19-s + (−0.170 − 0.467i)21-s + (0.895 + 1.06i)23-s + (0.499 + 2.83i)25-s + (0.0962 − 0.166i)27-s + (0.0149 − 0.0409i)29-s + (0.0944 + 0.163i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.748 - 0.663i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.748 - 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.748 - 0.663i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.748 - 0.663i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.846902 + 0.321153i\)
\(L(\frac12)\) \(\approx\) \(0.846902 + 0.321153i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.33 - 0.407i)T \)
good5 \( 1 + (3.37 + 2.82i)T + (0.868 + 4.92i)T^{2} \)
7 \( 1 + (1.97 - 1.14i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.618 - 0.357i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.61 - 0.285i)T + (12.2 - 4.44i)T^{2} \)
17 \( 1 + (-4.33 + 1.57i)T + (13.0 - 10.9i)T^{2} \)
23 \( 1 + (-4.29 - 5.12i)T + (-3.99 + 22.6i)T^{2} \)
29 \( 1 + (-0.0802 + 0.220i)T + (-22.2 - 18.6i)T^{2} \)
31 \( 1 + (-0.525 - 0.910i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 8.20iT - 37T^{2} \)
41 \( 1 + (-3.04 - 0.537i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (-3.07 + 3.66i)T + (-7.46 - 42.3i)T^{2} \)
47 \( 1 + (-2.76 + 7.59i)T + (-36.0 - 30.2i)T^{2} \)
53 \( 1 + (8.01 + 9.55i)T + (-9.20 + 52.1i)T^{2} \)
59 \( 1 + (-8.13 + 2.95i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-1.25 + 1.04i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (1.95 + 0.710i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-9.45 - 7.93i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (0.0741 - 0.420i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (2.98 - 16.9i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (8.29 - 4.78i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-9.14 + 1.61i)T + (83.6 - 30.4i)T^{2} \)
97 \( 1 + (0.956 + 2.62i)T + (-74.3 + 62.3i)T^{2} \)
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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

−9.862278903100811825651258117813, −9.425542678620197944629697795001, −8.570188951522687062235593437882, −7.77632927036858934381392724660, −6.96049029782471249273389873793, −5.42457460648152042949538407949, −4.98749924677122037971597581494, −3.82097829813550669217116849748, −3.16429490390782731637759386384, −0.941418509310169785371672474120, 0.61249475515057199937431070713, 2.78973812840867759189834442706, 3.40624461979813542861791190504, 4.43550407952945028980933912778, 5.94251769270975009223979359917, 6.79978191717531634447438004727, 7.46344368995273923126090437160, 7.890601783387428487785677491340, 9.135872371463534059437958497081, 10.24991246909775935782802989015

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