Properties

Label 2-30912-1.1-c1-0-2
Degree $2$
Conductor $30912$
Sign $1$
Analytic cond. $246.833$
Root an. cond. $15.7109$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 3·5-s − 7-s + 9-s − 4·11-s + 3·13-s + 3·15-s − 4·17-s + 21-s + 23-s + 4·25-s − 27-s − 3·29-s − 6·31-s + 4·33-s + 3·35-s + 9·37-s − 3·39-s + 9·41-s + 3·43-s − 3·45-s − 7·47-s + 49-s + 4·51-s + 4·53-s + 12·55-s − 6·59-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.34·5-s − 0.377·7-s + 1/3·9-s − 1.20·11-s + 0.832·13-s + 0.774·15-s − 0.970·17-s + 0.218·21-s + 0.208·23-s + 4/5·25-s − 0.192·27-s − 0.557·29-s − 1.07·31-s + 0.696·33-s + 0.507·35-s + 1.47·37-s − 0.480·39-s + 1.40·41-s + 0.457·43-s − 0.447·45-s − 1.02·47-s + 1/7·49-s + 0.560·51-s + 0.549·53-s + 1.61·55-s − 0.781·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 30912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(30912\)    =    \(2^{6} \cdot 3 \cdot 7 \cdot 23\)
Sign: $1$
Analytic conductor: \(246.833\)
Root analytic conductor: \(15.7109\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{30912} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 30912,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2823341462\)
\(L(\frac12)\) \(\approx\) \(0.2823341462\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 + T \)
23 \( 1 - T \)
good5 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 9 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 - 3 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−15.10451131098137, −14.93221998454065, −13.96437826065745, −13.29164801233636, −12.91651817729349, −12.54794702699033, −11.82644746485789, −11.13734467854866, −11.08762735293636, −10.52934046501132, −9.732580410822182, −9.089892353198202, −8.560638091290068, −7.787890602969249, −7.543068752102566, −6.932121036574722, −6.065873138480262, −5.786122962030576, −4.832654573728740, −4.364172300725274, −3.780078451966750, −3.072588236446446, −2.351126396534179, −1.247324784129891, −0.2251051074316618, 0.2251051074316618, 1.247324784129891, 2.351126396534179, 3.072588236446446, 3.780078451966750, 4.364172300725274, 4.832654573728740, 5.786122962030576, 6.065873138480262, 6.932121036574722, 7.543068752102566, 7.787890602969249, 8.560638091290068, 9.089892353198202, 9.732580410822182, 10.52934046501132, 11.08762735293636, 11.13734467854866, 11.82644746485789, 12.54794702699033, 12.91651817729349, 13.29164801233636, 13.96437826065745, 14.93221998454065, 15.10451131098137

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