Properties

Label 2-912-19.18-c2-0-10
Degree $2$
Conductor $912$
Sign $0.360 - 0.932i$
Analytic cond. $24.8502$
Root an. cond. $4.98499$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.73i·3-s + 2.94·5-s − 5.84·7-s − 2.99·9-s − 18.6·11-s − 1.55i·13-s − 5.10i·15-s + 12.7·17-s + (17.7 + 6.84i)19-s + 10.1i·21-s + 15.8·23-s − 16.2·25-s + 5.19i·27-s + 46.0i·29-s + 37.5i·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + 0.589·5-s − 0.835·7-s − 0.333·9-s − 1.69·11-s − 0.119i·13-s − 0.340i·15-s + 0.749·17-s + (0.932 + 0.360i)19-s + 0.482i·21-s + 0.687·23-s − 0.651·25-s + 0.192i·27-s + 1.58i·29-s + 1.21i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.360 - 0.932i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.360 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.360 - 0.932i$
Analytic conductor: \(24.8502\)
Root analytic conductor: \(4.98499\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (721, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1),\ 0.360 - 0.932i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.107664909\)
\(L(\frac12)\) \(\approx\) \(1.107664909\)
\(L(2)\) 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 \( 1 + 1.73iT \)
19 \( 1 + (-17.7 - 6.84i)T \)
good5 \( 1 - 2.94T + 25T^{2} \)
7 \( 1 + 5.84T + 49T^{2} \)
11 \( 1 + 18.6T + 121T^{2} \)
13 \( 1 + 1.55iT - 169T^{2} \)
17 \( 1 - 12.7T + 289T^{2} \)
23 \( 1 - 15.8T + 529T^{2} \)
29 \( 1 - 46.0iT - 841T^{2} \)
31 \( 1 - 37.5iT - 961T^{2} \)
37 \( 1 - 17.6iT - 1.36e3T^{2} \)
41 \( 1 - 33.3iT - 1.68e3T^{2} \)
43 \( 1 - 50.6T + 1.84e3T^{2} \)
47 \( 1 - 15.1T + 2.20e3T^{2} \)
53 \( 1 + 75.3iT - 2.80e3T^{2} \)
59 \( 1 - 53.0iT - 3.48e3T^{2} \)
61 \( 1 - 11.3T + 3.72e3T^{2} \)
67 \( 1 + 12.1iT - 4.48e3T^{2} \)
71 \( 1 - 20.8iT - 5.04e3T^{2} \)
73 \( 1 + 34.2T + 5.32e3T^{2} \)
79 \( 1 - 128. iT - 6.24e3T^{2} \)
83 \( 1 + 66.9T + 6.88e3T^{2} \)
89 \( 1 - 25.1iT - 7.92e3T^{2} \)
97 \( 1 - 117. iT - 9.40e3T^{2} \)
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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

−10.09152048209712903904690355546, −9.314961580011087804992939918169, −8.265169217732874181168020035729, −7.48576350624138623277036524621, −6.69439779353428746201447712713, −5.61674379131932199499477650494, −5.15079913571246437505910951374, −3.34623344166052365872066236342, −2.65376427303937302755677475059, −1.22239083627818058143543293004, 0.37371220281120373704279352200, 2.36739563575563567037965125894, 3.17356890826312993092056547266, 4.39518085607933548803870894048, 5.56124763724962394387509679728, 5.91536950450946796324214141110, 7.32413286336816943957079866581, 7.977479571735440967835673613707, 9.240782234812836039969602606080, 9.738438768851209396217210416591

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