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Properties

Label 2-960-1.1-c1-0-14
Degree $2$
Conductor $960$
Sign $-1$
Analytic cond. $7.66563$
Root an. cond. $2.76868$
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 − 5-s + 9-s − 4·11-s − 6·13-s − 15-s − 6·17-s − 4·19-s + 25-s + 27-s + 2·29-s + 8·31-s − 4·33-s + 2·37-s − 6·39-s − 6·41-s + 12·43-s − 45-s − 8·47-s − 7·49-s − 6·51-s − 6·53-s + 4·55-s − 4·57-s + 12·59-s − 14·61-s + 6·65-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.20·11-s − 1.66·13-s − 0.258·15-s − 1.45·17-s − 0.917·19-s + 1/5·25-s + 0.192·27-s + 0.371·29-s + 1.43·31-s − 0.696·33-s + 0.328·37-s − 0.960·39-s − 0.937·41-s + 1.82·43-s − 0.149·45-s − 1.16·47-s − 49-s − 0.840·51-s − 0.824·53-s + 0.539·55-s − 0.529·57-s + 1.56·59-s − 1.79·61-s + 0.744·65-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $-1$
Analytic conductor: \(7.66563\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 960,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
5 \( 1 + T \)
good7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
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 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−9.647859917826187558725505229428, −8.645810177900386367635479957201, −7.982386705736041440880055703756, −7.23083008412003813869292936859, −6.33640038439988253692354509575, −4.88793662384328778723790668153, −4.41954513526927827595737015950, −2.92311775669320789657547583777, −2.22526038506206736624914155682, 0, 2.22526038506206736624914155682, 2.92311775669320789657547583777, 4.41954513526927827595737015950, 4.88793662384328778723790668153, 6.33640038439988253692354509575, 7.23083008412003813869292936859, 7.982386705736041440880055703756, 8.645810177900386367635479957201, 9.647859917826187558725505229428

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