Properties

Label 2-912-57.8-c1-0-18
Degree $2$
Conductor $912$
Sign $0.992 - 0.120i$
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  + (1.57 − 0.724i)3-s + (1.22 + 0.707i)5-s + 0.267·7-s + (1.94 − 2.28i)9-s + 5.27i·11-s + (−0.232 + 0.133i)13-s + (2.43 + 0.224i)15-s + (4.24 + 2.44i)17-s + (−1.73 + 4i)19-s + (0.421 − 0.194i)21-s + (4.57 − 2.63i)23-s + (−1.50 − 2.59i)25-s + (1.41 − 5.00i)27-s + (1.03 + 1.79i)29-s − 2.46i·31-s + ⋯
L(s)  = 1  + (0.908 − 0.418i)3-s + (0.547 + 0.316i)5-s + 0.101·7-s + (0.649 − 0.760i)9-s + 1.59i·11-s + (−0.0643 + 0.0371i)13-s + (0.629 + 0.0580i)15-s + (1.02 + 0.594i)17-s + (−0.397 + 0.917i)19-s + (0.0919 − 0.0423i)21-s + (0.953 − 0.550i)23-s + (−0.300 − 0.519i)25-s + (0.272 − 0.962i)27-s + (0.192 + 0.332i)29-s − 0.442i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.120i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.992 - 0.120i$
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: \((2,\ 912,\ (\ :1/2),\ 0.992 - 0.120i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.45410 + 0.148620i\)
\(L(\frac12)\) \(\approx\) \(2.45410 + 0.148620i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.57 + 0.724i)T \)
19 \( 1 + (1.73 - 4i)T \)
good5 \( 1 + (-1.22 - 0.707i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 - 0.267T + 7T^{2} \)
11 \( 1 - 5.27iT - 11T^{2} \)
13 \( 1 + (0.232 - 0.133i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-4.24 - 2.44i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-4.57 + 2.63i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.03 - 1.79i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.46iT - 31T^{2} \)
37 \( 1 + 7.73iT - 37T^{2} \)
41 \( 1 + (2.82 - 4.89i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.86 + 4.96i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.656 + 0.378i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.46 + 9.46i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.60 - 9.71i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.23 - 9.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6.69 + 11.5i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.5 + 2.59i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (9.06 + 5.23i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.07iT - 83T^{2} \)
89 \( 1 + (-3.67 - 6.36i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.464 + 0.267i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.989806266929107579862466424944, −9.369595837637582095702968598426, −8.352234331672157124279951521365, −7.59421506736130961115441457131, −6.84580218528106452646758849184, −5.95005323001668172215384439535, −4.66315205620051863385202513106, −3.64093170052076480974374695206, −2.42401625082568662914241303228, −1.59237011117653558162478786126, 1.26208926721434338163630284158, 2.79564905164361778775530653270, 3.46941770451879592166586638229, 4.82883385406660170728445990119, 5.53194687821988320707875493814, 6.69990489361205828565400423504, 7.80242218373494912758222447382, 8.494864815700259997744010161777, 9.269521333439895806402051770224, 9.812383101093650997201855976077

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