Properties

Label 2-912-57.8-c1-0-12
Degree $2$
Conductor $912$
Sign $0.974 + 0.223i$
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 + 1.41i)3-s + (−1.22 − 0.707i)5-s − 4.44·7-s + (−1.00 − 2.82i)9-s + 0.317i·11-s + (−3 + 1.73i)13-s + (2.22 − 1.02i)15-s + (5.44 + 3.14i)17-s + (4.17 − 1.25i)19-s + (4.44 − 6.29i)21-s + (6.12 − 3.53i)23-s + (−1.50 − 2.59i)25-s + (5.00 + 1.41i)27-s + (1.22 + 2.12i)29-s − 4.24i·31-s + ⋯
L(s)  = 1  + (−0.577 + 0.816i)3-s + (−0.547 − 0.316i)5-s − 1.68·7-s + (−0.333 − 0.942i)9-s + 0.0958i·11-s + (−0.832 + 0.480i)13-s + (0.574 − 0.264i)15-s + (1.32 + 0.763i)17-s + (0.957 − 0.287i)19-s + (0.970 − 1.37i)21-s + (1.27 − 0.737i)23-s + (−0.300 − 0.519i)25-s + (0.962 + 0.272i)27-s + (0.227 + 0.393i)29-s − 0.762i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 + 0.223i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 + 0.223i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.974 + 0.223i$
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: \((2,\ 912,\ (\ :1/2),\ 0.974 + 0.223i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.737757 - 0.0835237i\)
\(L(\frac12)\) \(\approx\) \(0.737757 - 0.0835237i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - 1.41i)T \)
19 \( 1 + (-4.17 + 1.25i)T \)
good5 \( 1 + (1.22 + 0.707i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + 4.44T + 7T^{2} \)
11 \( 1 - 0.317iT - 11T^{2} \)
13 \( 1 + (3 - 1.73i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-5.44 - 3.14i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-6.12 + 3.53i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.22 - 2.12i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4.24iT - 31T^{2} \)
37 \( 1 + 0.778iT - 37T^{2} \)
41 \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.449 + 0.778i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.57 - 3.21i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.550 - 0.953i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.27 + 5.67i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.22 - 5.58i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.17 + 2.98i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3 + 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.39 + 9.35i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.34 + 4.24i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 14.1iT - 83T^{2} \)
89 \( 1 + (8.44 + 14.6i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (11.8 + 6.84i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.836959773301188898734791113189, −9.587482360145861487947927988515, −8.599156119294980781457460622198, −7.37206463465624849193679809174, −6.54991283251128320069479648102, −5.67740555386351421032890242367, −4.71287191328126203410730842914, −3.71998252168772947833781714596, −2.94692178253096914489981787084, −0.54173279679225945125661524937, 0.878485682299211605384982184465, 2.85249934357193251255091010165, 3.41827953563966253234003387937, 5.15817441501969198750228858504, 5.79255406202675722670948706982, 7.03758604811767970836758870691, 7.19423718259712107103448009408, 8.203793115366558098800662561131, 9.610944495353858349946298120809, 9.942147618805559231873050984762

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