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Properties

Label 2-912-228.215-c1-0-5
Degree $2$
Conductor $912$
Sign $-0.991 - 0.129i$
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  + (−0.592 + 1.62i)3-s + (1.63 + 0.943i)7-s + (−2.29 − 1.92i)9-s + (−6.08 + 2.21i)13-s + (−0.5 + 4.33i)19-s + (−2.50 + 2.10i)21-s + (−4.69 + 1.71i)25-s + (4.5 − 2.59i)27-s + (4.66 + 2.69i)31-s − 3.20·37-s − 11.2i·39-s + (−12.9 + 2.27i)43-s + (−1.71 − 2.97i)49-s + (−6.75 − 3.37i)57-s + (−0.762 + 4.32i)61-s + ⋯
L(s)  = 1  + (−0.342 + 0.939i)3-s + (0.617 + 0.356i)7-s + (−0.766 − 0.642i)9-s + (−1.68 + 0.614i)13-s + (−0.114 + 0.993i)19-s + (−0.546 + 0.458i)21-s + (−0.939 + 0.342i)25-s + (0.866 − 0.499i)27-s + (0.838 + 0.484i)31-s − 0.527·37-s − 1.79i·39-s + (−1.96 + 0.346i)43-s + (−0.245 − 0.425i)49-s + (−0.894 − 0.447i)57-s + (−0.0976 + 0.553i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.129i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.129i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.991 - 0.129i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (671, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.991 - 0.129i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0455479 + 0.699137i\)
\(L(\frac12)\) \(\approx\) \(0.0455479 + 0.699137i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.592 - 1.62i)T \)
19 \( 1 + (0.5 - 4.33i)T \)
good5 \( 1 + (4.69 - 1.71i)T^{2} \)
7 \( 1 + (-1.63 - 0.943i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (6.08 - 2.21i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (-2.95 + 16.7i)T^{2} \)
23 \( 1 + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (-5.03 - 28.5i)T^{2} \)
31 \( 1 + (-4.66 - 2.69i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 3.20T + 37T^{2} \)
41 \( 1 + (-31.4 - 26.3i)T^{2} \)
43 \( 1 + (12.9 - 2.27i)T + (40.4 - 14.7i)T^{2} \)
47 \( 1 + (8.16 + 46.2i)T^{2} \)
53 \( 1 + (49.8 + 18.1i)T^{2} \)
59 \( 1 + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (0.762 - 4.32i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-1.42 + 1.69i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (14.4 + 5.26i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (5.09 - 14.0i)T + (-60.5 - 50.7i)T^{2} \)
83 \( 1 + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-68.1 + 57.2i)T^{2} \)
97 \( 1 + (-14.5 + 12.2i)T + (16.8 - 95.5i)T^{2} \)
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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

−10.16352510462839174249652829619, −9.943128470987361861770344572843, −8.913420008215262274449393610519, −8.118940482518010667523702040609, −7.08162784268925965696350957488, −5.99246028745763803769032047268, −5.07831340279622912027278902826, −4.47319995445359238366800532126, −3.29135045865570922281486033021, −1.96513582016621737070182723937, 0.32589764396672231637462322538, 1.88871243830551234459117170696, 2.91336709552213079249836290300, 4.59118378753662316257461448643, 5.25634109765324018923881494419, 6.35634755325490804292787892146, 7.26773525914116564781467947232, 7.79343663371102897794041158302, 8.638106879942035239595369428270, 9.838455636403342876981847841219

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