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Properties

Label 2-912-57.50-c1-0-9
Degree $2$
Conductor $912$
Sign $-0.0799 - 0.996i$
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  + (1.68 + 0.396i)3-s + (−0.813 + 0.469i)5-s − 3.37·7-s + (2.68 + 1.33i)9-s + 5.04i·11-s + (1.5 + 0.866i)13-s + (−1.55 + 0.469i)15-s + (5.18 − 2.99i)17-s + (−4 + 1.73i)19-s + (−5.68 − 1.33i)21-s + (−0.813 − 0.469i)23-s + (−2.05 + 3.56i)25-s + (4 + 3.31i)27-s + (−5.18 + 8.98i)29-s + 2.37i·31-s + ⋯
L(s)  = 1  + (0.973 + 0.228i)3-s + (−0.363 + 0.210i)5-s − 1.27·7-s + (0.895 + 0.445i)9-s + 1.52i·11-s + (0.416 + 0.240i)13-s + (−0.402 + 0.121i)15-s + (1.25 − 0.726i)17-s + (−0.917 + 0.397i)19-s + (−1.24 − 0.291i)21-s + (−0.169 − 0.0979i)23-s + (−0.411 + 0.713i)25-s + (0.769 + 0.638i)27-s + (−0.963 + 1.66i)29-s + 0.426i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0799 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0799 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.0799 - 0.996i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.0799 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09044 + 1.18135i\)
\(L(\frac12)\) \(\approx\) \(1.09044 + 1.18135i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.68 - 0.396i)T \)
19 \( 1 + (4 - 1.73i)T \)
good5 \( 1 + (0.813 - 0.469i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + 3.37T + 7T^{2} \)
11 \( 1 - 5.04iT - 11T^{2} \)
13 \( 1 + (-1.5 - 0.866i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-5.18 + 2.99i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (0.813 + 0.469i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (5.18 - 8.98i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.37iT - 31T^{2} \)
37 \( 1 + 5.84iT - 37T^{2} \)
41 \( 1 + (-3.55 - 6.16i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.872 - 1.51i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.18 - 2.99i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.813 + 1.40i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.813 + 1.40i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.872 - 1.51i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.5 - 0.866i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.55 + 6.16i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (6.87 + 11.9i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-8.61 + 4.97i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.87iT - 83T^{2} \)
89 \( 1 + (-0.813 + 1.40i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-10.6 + 6.16i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.07192444798873145946595471446, −9.477417505437007682125120383024, −8.861060026538715494977270065462, −7.51350891610269822666519676326, −7.28932416372309826516161752877, −6.13584368601628266481570881960, −4.80739949336608953990795207398, −3.76257592193273816702406828259, −3.10469904439605604485754734023, −1.81568769538916993568357221426, 0.67744491821515293500821410556, 2.46769375541489776586627723733, 3.52969877195087971152043969436, 3.99127848101656112628571897350, 5.83872714084213988180374949686, 6.32764471902106039244084474650, 7.56022208401804043914830358868, 8.273026883761412669923682270846, 8.884448533867363432285958979698, 9.811571029429735751466829147664

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