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Properties

Label 2-912-57.8-c1-0-10
Degree $2$
Conductor $912$
Sign $-0.530 - 0.847i$
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  + (1.13 + 1.30i)3-s + (1.87 + 1.08i)5-s − 3.41·7-s + (−0.420 + 2.97i)9-s − 2.39i·11-s + (−4.61 + 2.66i)13-s + (0.712 + 3.67i)15-s + (5.15 + 2.97i)17-s + (−1.09 + 4.21i)19-s + (−3.87 − 4.46i)21-s + (−3.01 + 1.74i)23-s + (−0.159 − 0.276i)25-s + (−4.36 + 2.82i)27-s + (4.68 + 8.10i)29-s + 4.72i·31-s + ⋯
L(s)  = 1  + (0.655 + 0.755i)3-s + (0.837 + 0.483i)5-s − 1.28·7-s + (−0.140 + 0.990i)9-s − 0.723i·11-s + (−1.27 + 0.738i)13-s + (0.184 + 0.949i)15-s + (1.25 + 0.722i)17-s + (−0.251 + 0.967i)19-s + (−0.845 − 0.973i)21-s + (−0.628 + 0.362i)23-s + (−0.0319 − 0.0553i)25-s + (−0.839 + 0.543i)27-s + (0.869 + 1.50i)29-s + 0.848i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.530 - 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.530 - 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.530 - 0.847i$
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: \((2,\ 912,\ (\ :1/2),\ -0.530 - 0.847i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.788789 + 1.42383i\)
\(L(\frac12)\) \(\approx\) \(0.788789 + 1.42383i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.13 - 1.30i)T \)
19 \( 1 + (1.09 - 4.21i)T \)
good5 \( 1 + (-1.87 - 1.08i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + 3.41T + 7T^{2} \)
11 \( 1 + 2.39iT - 11T^{2} \)
13 \( 1 + (4.61 - 2.66i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-5.15 - 2.97i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (3.01 - 1.74i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.68 - 8.10i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.72iT - 31T^{2} \)
37 \( 1 - 1.54iT - 37T^{2} \)
41 \( 1 + (-4.63 + 8.02i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.248 - 0.430i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-7.18 + 4.15i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.46 + 7.73i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.192 - 0.332i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.38 + 2.40i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.99 + 2.88i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.344 - 0.596i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.06 + 7.04i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.09 - 3.52i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 1.73iT - 83T^{2} \)
89 \( 1 + (-8.86 - 15.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (12.3 + 7.12i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.14914546686174090283312517226, −9.726777109061489870592072542903, −8.934124840874067494833451516225, −7.969601156380761197930324490639, −6.89720536659383235550208430515, −6.03164870576251913264314290972, −5.19397858894476352040273232803, −3.78085898819556899825223517429, −3.12517952508770332682248268122, −2.04167728827917072354102794861, 0.67670379571129808199561622510, 2.35522607758045953183398167411, 2.93874439741883081409931272943, 4.42530644164214894434551435063, 5.65323096945262427022864696273, 6.39033241239084623088246145669, 7.36047610582143162988710717775, 7.955293738238734788146634861262, 9.308849546876944101813320840497, 9.611174520685025741825486530756

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