Properties

Label 2-30912-1.1-c1-0-5
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 − 2·5-s + 7-s + 9-s − 5·11-s − 4·13-s + 2·15-s + 6·17-s + 5·19-s − 21-s + 23-s − 25-s − 27-s + 2·31-s + 5·33-s − 2·35-s + 8·37-s + 4·39-s + 3·41-s − 2·45-s + 11·47-s + 49-s − 6·51-s − 53-s + 10·55-s − 5·57-s − 13·59-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s − 1.50·11-s − 1.10·13-s + 0.516·15-s + 1.45·17-s + 1.14·19-s − 0.218·21-s + 0.208·23-s − 1/5·25-s − 0.192·27-s + 0.359·31-s + 0.870·33-s − 0.338·35-s + 1.31·37-s + 0.640·39-s + 0.468·41-s − 0.298·45-s + 1.60·47-s + 1/7·49-s − 0.840·51-s − 0.137·53-s + 1.34·55-s − 0.662·57-s − 1.69·59-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

\(L(1)\) \(\approx\) \(1.118771298\)
\(L(\frac12)\) \(\approx\) \(1.118771298\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 11 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 + 13 T + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−15.06596961130830, −14.77862959346538, −14.00197016708854, −13.56436160951509, −12.77023037910774, −12.33506358903849, −11.97806645975118, −11.41540549466023, −10.88683494533352, −10.32621122622936, −9.775780852821733, −9.362698457729367, −8.276842015866682, −7.864183602591667, −7.478702930647339, −7.140005286142886, −6.017028115777436, −5.515319917908156, −5.037969332897399, −4.484697878069095, −3.726681605795537, −2.923006474565290, −2.417546729617288, −1.210108662140008, −0.4648807968069634, 0.4648807968069634, 1.210108662140008, 2.417546729617288, 2.923006474565290, 3.726681605795537, 4.484697878069095, 5.037969332897399, 5.515319917908156, 6.017028115777436, 7.140005286142886, 7.478702930647339, 7.864183602591667, 8.276842015866682, 9.362698457729367, 9.775780852821733, 10.32621122622936, 10.88683494533352, 11.41540549466023, 11.97806645975118, 12.33506358903849, 12.77023037910774, 13.56436160951509, 14.00197016708854, 14.77862959346538, 15.06596961130830

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