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Properties

Label 2-912-228.11-c1-0-20
Degree $2$
Conductor $912$
Sign $0.954 + 0.296i$
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  + (−0.291 − 1.70i)3-s + (3.50 + 2.02i)5-s + 0.196i·7-s + (−2.83 + 0.994i)9-s + 0.452·11-s + (−0.564 − 0.978i)13-s + (2.43 − 6.58i)15-s + (5.05 + 2.91i)17-s + (2.75 − 3.38i)19-s + (0.335 − 0.0572i)21-s + (−0.987 − 1.70i)23-s + (5.71 + 9.89i)25-s + (2.52 + 4.54i)27-s + (4.05 − 2.33i)29-s + 6.15i·31-s + ⋯
L(s)  = 1  + (−0.168 − 0.985i)3-s + (1.56 + 0.906i)5-s + 0.0743i·7-s + (−0.943 + 0.331i)9-s + 0.136·11-s + (−0.156 − 0.271i)13-s + (0.629 − 1.69i)15-s + (1.22 + 0.707i)17-s + (0.631 − 0.775i)19-s + (0.0732 − 0.0125i)21-s + (−0.205 − 0.356i)23-s + (1.14 + 1.97i)25-s + (0.485 + 0.874i)27-s + (0.752 − 0.434i)29-s + 1.10i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 + 0.296i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.954 + 0.296i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.954 + 0.296i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.954 + 0.296i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.96558 - 0.298594i\)
\(L(\frac12)\) \(\approx\) \(1.96558 - 0.298594i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.291 + 1.70i)T \)
19 \( 1 + (-2.75 + 3.38i)T \)
good5 \( 1 + (-3.50 - 2.02i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 - 0.196iT - 7T^{2} \)
11 \( 1 - 0.452T + 11T^{2} \)
13 \( 1 + (0.564 + 0.978i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-5.05 - 2.91i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (0.987 + 1.70i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.05 + 2.33i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.15iT - 31T^{2} \)
37 \( 1 + 6.61T + 37T^{2} \)
41 \( 1 + (5.10 + 2.94i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-9.75 - 5.63i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.48 + 7.76i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.88 + 2.81i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.360 + 0.624i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.652 - 1.13i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.13 - 4.69i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.53 + 4.38i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.644 + 1.11i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.36 + 2.51i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 11.1T + 83T^{2} \)
89 \( 1 + (3.32 - 1.91i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.06 + 12.2i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.25934450705817158657988624110, −9.256069231582782485620458916586, −8.332598570223097933286278315022, −7.23903915253771579936548892299, −6.64862366035452898334750419410, −5.81436528448205139847198888753, −5.22493245857907408580532512332, −3.24372876844750880177265213891, −2.39703584763974754382485988705, −1.31962653457230464105068849622, 1.21772757083653275991590288775, 2.65593678162223266565433555124, 3.94020230290772688352322308281, 5.10498600226288324028700785140, 5.51548108535833585740760962090, 6.35494581863623282068308565623, 7.74847123717918627886081010780, 8.850907514606992330613743219836, 9.435888642483216072102982928615, 9.981873641235902136018526612988

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