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Properties

Label 2-912-228.107-c0-0-1
Degree $2$
Conductor $912$
Sign $0.910 - 0.412i$
Analytic cond. $0.455147$
Root an. cond. $0.674646$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)3-s − 1.73i·7-s + (−0.499 + 0.866i)9-s + (1.5 + 0.866i)13-s + 19-s + (1.49 − 0.866i)21-s + (−0.5 + 0.866i)25-s − 0.999·27-s − 31-s − 1.73i·37-s + 1.73i·39-s + (−1.5 + 0.866i)43-s − 1.99·49-s + (0.5 + 0.866i)57-s + (−0.5 + 0.866i)61-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s − 1.73i·7-s + (−0.499 + 0.866i)9-s + (1.5 + 0.866i)13-s + 19-s + (1.49 − 0.866i)21-s + (−0.5 + 0.866i)25-s − 0.999·27-s − 31-s − 1.73i·37-s + 1.73i·39-s + (−1.5 + 0.866i)43-s − 1.99·49-s + (0.5 + 0.866i)57-s + (−0.5 + 0.866i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.910 - 0.412i$
Analytic conductor: \(0.455147\)
Root analytic conductor: \(0.674646\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (335, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :0),\ 0.910 - 0.412i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.205577939\)
\(L(\frac12)\) \(\approx\) \(1.205577939\)
\(L(1)\) 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 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 - T \)
good5 \( 1 + (0.5 - 0.866i)T^{2} \)
7 \( 1 + 1.73iT - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 + 1.73iT - T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−10.36352088995836544751663582398, −9.519992304093079476585738558546, −8.861772297385606034400995761666, −7.78596569086055731794282189680, −7.17143060953787714839565598715, −5.99165484588640809411999551258, −4.82640766478715929900894341162, −3.84920861665654669624569662995, −3.45640035068502334256842326387, −1.57213550181728894720236107347, 1.55534855442202624401662324813, 2.73948225266767511004073887190, 3.53606581861919412649860763948, 5.31315066045061950650358004513, 5.94050629678203828923026731400, 6.74209416609971380991848882778, 8.022366461424400834894634235743, 8.440459865224329581018570531611, 9.147564389348741810055681283591, 10.08354571323028296743835279195

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