Properties

Label 2-38976-1.1-c1-0-50
Degree $2$
Conductor $38976$
Sign $-1$
Analytic cond. $311.224$
Root an. cond. $17.6415$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 2·5-s + 7-s + 9-s − 4·11-s + 2·13-s + 2·15-s + 2·17-s + 4·19-s + 21-s − 25-s + 27-s − 29-s − 8·31-s − 4·33-s + 2·35-s + 10·37-s + 2·39-s − 6·41-s − 12·43-s + 2·45-s − 8·47-s + 49-s + 2·51-s − 6·53-s − 8·55-s + 4·57-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s + 0.377·7-s + 1/3·9-s − 1.20·11-s + 0.554·13-s + 0.516·15-s + 0.485·17-s + 0.917·19-s + 0.218·21-s − 1/5·25-s + 0.192·27-s − 0.185·29-s − 1.43·31-s − 0.696·33-s + 0.338·35-s + 1.64·37-s + 0.320·39-s − 0.937·41-s − 1.82·43-s + 0.298·45-s − 1.16·47-s + 1/7·49-s + 0.280·51-s − 0.824·53-s − 1.07·55-s + 0.529·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38976 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38976 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(38976\)    =    \(2^{6} \cdot 3 \cdot 7 \cdot 29\)
Sign: $-1$
Analytic conductor: \(311.224\)
Root analytic conductor: \(17.6415\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 38976,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
29 \( 1 + T \)
good5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−15.00637725139277, −14.53377201858604, −13.95589180660318, −13.55193609754475, −13.06092910076387, −12.77218427231595, −11.94034627890357, −11.25462643527992, −10.96454431447113, −10.04041560737664, −9.878147791673020, −9.387548033174583, −8.545301081183040, −8.212586996231548, −7.587224943398898, −7.143567357089676, −6.260663723622286, −5.771351463602264, −5.152427405614522, −4.753691744233215, −3.709914943625801, −3.220134746524246, −2.534331491517699, −1.794143681589079, −1.296026765886334, 0, 1.296026765886334, 1.794143681589079, 2.534331491517699, 3.220134746524246, 3.709914943625801, 4.753691744233215, 5.152427405614522, 5.771351463602264, 6.260663723622286, 7.143567357089676, 7.587224943398898, 8.212586996231548, 8.545301081183040, 9.387548033174583, 9.878147791673020, 10.04041560737664, 10.96454431447113, 11.25462643527992, 11.94034627890357, 12.77218427231595, 13.06092910076387, 13.55193609754475, 13.95589180660318, 14.53377201858604, 15.00637725139277

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