Properties

Label 2-960-1.1-c3-0-43
Degree $2$
Conductor $960$
Sign $-1$
Analytic cond. $56.6418$
Root an. cond. $7.52607$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 5·5-s − 20·7-s + 9·9-s + 16·11-s − 58·13-s + 15·15-s + 38·17-s + 4·19-s − 60·21-s + 80·23-s + 25·25-s + 27·27-s − 82·29-s + 8·31-s + 48·33-s − 100·35-s − 426·37-s − 174·39-s − 246·41-s − 524·43-s + 45·45-s + 464·47-s + 57·49-s + 114·51-s + 702·53-s + 80·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 1.07·7-s + 1/3·9-s + 0.438·11-s − 1.23·13-s + 0.258·15-s + 0.542·17-s + 0.0482·19-s − 0.623·21-s + 0.725·23-s + 1/5·25-s + 0.192·27-s − 0.525·29-s + 0.0463·31-s + 0.253·33-s − 0.482·35-s − 1.89·37-s − 0.714·39-s − 0.937·41-s − 1.85·43-s + 0.149·45-s + 1.44·47-s + 0.166·49-s + 0.313·51-s + 1.81·53-s + 0.196·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $-1$
Analytic conductor: \(56.6418\)
Root analytic conductor: \(7.52607\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 960,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 - p T \)
5 \( 1 - p T \)
good7 \( 1 + 20 T + p^{3} T^{2} \)
11 \( 1 - 16 T + p^{3} T^{2} \)
13 \( 1 + 58 T + p^{3} T^{2} \)
17 \( 1 - 38 T + p^{3} T^{2} \)
19 \( 1 - 4 T + p^{3} T^{2} \)
23 \( 1 - 80 T + p^{3} T^{2} \)
29 \( 1 + 82 T + p^{3} T^{2} \)
31 \( 1 - 8 T + p^{3} T^{2} \)
37 \( 1 + 426 T + p^{3} T^{2} \)
41 \( 1 + 6 p T + p^{3} T^{2} \)
43 \( 1 + 524 T + p^{3} T^{2} \)
47 \( 1 - 464 T + p^{3} T^{2} \)
53 \( 1 - 702 T + p^{3} T^{2} \)
59 \( 1 + 592 T + p^{3} T^{2} \)
61 \( 1 + 574 T + p^{3} T^{2} \)
67 \( 1 + 172 T + p^{3} T^{2} \)
71 \( 1 + 768 T + p^{3} T^{2} \)
73 \( 1 + 558 T + p^{3} T^{2} \)
79 \( 1 + 408 T + p^{3} T^{2} \)
83 \( 1 - 164 T + p^{3} T^{2} \)
89 \( 1 + 510 T + p^{3} T^{2} \)
97 \( 1 - 514 T + p^{3} T^{2} \)
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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

−9.258861611092421084996934476276, −8.615449786043061846049111505009, −7.33402162989635761896162706480, −6.87319282785099351875518777592, −5.78077557512811361395963803430, −4.81699506521133015431295765250, −3.55106538517011649759954752000, −2.81555554931744393922986571996, −1.61138413296123047322237239267, 0, 1.61138413296123047322237239267, 2.81555554931744393922986571996, 3.55106538517011649759954752000, 4.81699506521133015431295765250, 5.78077557512811361395963803430, 6.87319282785099351875518777592, 7.33402162989635761896162706480, 8.615449786043061846049111505009, 9.258861611092421084996934476276

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