Properties

Label 2-912-57.8-c1-0-21
Degree $2$
Conductor $912$
Sign $0.740 - 0.671i$
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  + (1.5 + 0.866i)3-s + 5·7-s + (1.5 + 2.59i)9-s + (−4.5 + 2.59i)13-s + (4 + 1.73i)19-s + (7.5 + 4.33i)21-s + (−2.5 − 4.33i)25-s + 5.19i·27-s − 8.66i·31-s − 5.19i·37-s − 9·39-s + (−6.5 + 11.2i)43-s + 18·49-s + (4.5 + 6.06i)57-s + (−0.5 − 0.866i)61-s + ⋯
L(s)  = 1  + (0.866 + 0.499i)3-s + 1.88·7-s + (0.5 + 0.866i)9-s + (−1.24 + 0.720i)13-s + (0.917 + 0.397i)19-s + (1.63 + 0.944i)21-s + (−0.5 − 0.866i)25-s + 0.999i·27-s − 1.55i·31-s − 0.854i·37-s − 1.44·39-s + (−0.991 + 1.71i)43-s + 2.57·49-s + (0.596 + 0.802i)57-s + (−0.0640 − 0.110i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.740 - 0.671i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.740 - 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.740 - 0.671i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.740 - 0.671i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.33318 + 0.900197i\)
\(L(\frac12)\) \(\approx\) \(2.33318 + 0.900197i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 - 0.866i)T \)
19 \( 1 + (-4 - 1.73i)T \)
good5 \( 1 + (2.5 + 4.33i)T^{2} \)
7 \( 1 - 5T + 7T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + (4.5 - 2.59i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 8.66iT - 31T^{2} \)
37 \( 1 + 5.19iT - 37T^{2} \)
41 \( 1 + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.5 - 11.2i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-10.5 + 6.06i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (8.5 - 14.7i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.5 - 2.59i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (12 + 6.92i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.945658478946246537714761087508, −9.469820562591393852898259193574, −8.333419746413785022011167173882, −7.88813762640433061363143145341, −7.14651385057017684132040019813, −5.54965912198979869561048570285, −4.68743780324641005896945955857, −4.09801939102367821626202153730, −2.56707010043686208432243447665, −1.70088139618554623603763509828, 1.28526050084237823454170558749, 2.30585572546382674002330257810, 3.45352670381144064493442280730, 4.80394077382316967124084758876, 5.36620250397231844451617001840, 6.98042654099905501436006492195, 7.56938337420508142754658130787, 8.216998760458442233424620998517, 8.957020556546414183889186123776, 9.928612178167159912809993714179

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