Properties

Label 2-102960-1.1-c1-0-2
Degree $2$
Conductor $102960$
Sign $1$
Analytic cond. $822.139$
Root an. cond. $28.6729$
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  − 5-s − 4·7-s − 11-s − 13-s + 17-s + 19-s − 6·23-s + 25-s − 8·29-s + 2·31-s + 4·35-s + 5·37-s + 5·41-s − 5·43-s + 9·47-s + 9·49-s + 55-s + 6·59-s − 2·61-s + 65-s + 3·67-s + 10·71-s + 16·73-s + 4·77-s − 14·79-s + 2·83-s − 85-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.51·7-s − 0.301·11-s − 0.277·13-s + 0.242·17-s + 0.229·19-s − 1.25·23-s + 1/5·25-s − 1.48·29-s + 0.359·31-s + 0.676·35-s + 0.821·37-s + 0.780·41-s − 0.762·43-s + 1.31·47-s + 9/7·49-s + 0.134·55-s + 0.781·59-s − 0.256·61-s + 0.124·65-s + 0.366·67-s + 1.18·71-s + 1.87·73-s + 0.455·77-s − 1.57·79-s + 0.219·83-s − 0.108·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(102960\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 11 \cdot 13\)
Sign: $1$
Analytic conductor: \(822.139\)
Root analytic conductor: \(28.6729\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 102960,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6523950353\)
\(L(\frac12)\) \(\approx\) \(0.6523950353\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + T \)
11 \( 1 + T \)
13 \( 1 + T \)
good7 \( 1 + 4 T + p T^{2} \)
17 \( 1 - T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
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

−13.58394150116102, −13.26030968799960, −12.56615029873638, −12.48976265109341, −11.82769100701042, −11.30494046880458, −10.79186960547406, −10.15631389311420, −9.751859144317212, −9.429187274403771, −8.852492704479313, −8.100227873313385, −7.781741205923418, −7.143587178158525, −6.722365219632868, −6.082744849595022, −5.667448864308764, −5.111523067135272, −4.191862714620916, −3.865598435176884, −3.318348899458966, −2.624572757349419, −2.168806414926272, −1.094324603426468, −0.2762053371312800, 0.2762053371312800, 1.094324603426468, 2.168806414926272, 2.624572757349419, 3.318348899458966, 3.865598435176884, 4.191862714620916, 5.111523067135272, 5.667448864308764, 6.082744849595022, 6.722365219632868, 7.143587178158525, 7.781741205923418, 8.100227873313385, 8.852492704479313, 9.429187274403771, 9.751859144317212, 10.15631389311420, 10.79186960547406, 11.30494046880458, 11.82769100701042, 12.48976265109341, 12.56615029873638, 13.26030968799960, 13.58394150116102

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