Properties

Label 2-30960-1.1-c1-0-7
Degree $2$
Conductor $30960$
Sign $1$
Analytic cond. $247.216$
Root an. cond. $15.7231$
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 − 3·11-s + 5·13-s + 7·17-s − 9·23-s + 25-s − 7·31-s − 4·35-s − 8·37-s − 3·41-s + 43-s − 8·47-s + 9·49-s + 53-s + 3·55-s − 8·59-s − 5·65-s + 9·67-s − 12·71-s − 4·73-s − 12·77-s + 8·79-s + 3·83-s − 7·85-s + 8·89-s + 20·91-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.51·7-s − 0.904·11-s + 1.38·13-s + 1.69·17-s − 1.87·23-s + 1/5·25-s − 1.25·31-s − 0.676·35-s − 1.31·37-s − 0.468·41-s + 0.152·43-s − 1.16·47-s + 9/7·49-s + 0.137·53-s + 0.404·55-s − 1.04·59-s − 0.620·65-s + 1.09·67-s − 1.42·71-s − 0.468·73-s − 1.36·77-s + 0.900·79-s + 0.329·83-s − 0.759·85-s + 0.847·89-s + 2.09·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(30960\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 43\)
Sign: $1$
Analytic conductor: \(247.216\)
Root analytic conductor: \(15.7231\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{30960} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 30960,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.387988866\)
\(L(\frac12)\) \(\approx\) \(2.387988866\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
43 \( 1 - T \)
good7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 9 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
97 \( 1 - 9 T + p T^{2} \)
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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

−15.00287600118620, −14.59703381938023, −13.99892149620467, −13.74085118006921, −12.94697228382210, −12.34736839437848, −11.85383203732188, −11.41518617392460, −10.79606754031375, −10.42470824940304, −9.858604396618121, −8.978547120924256, −8.356016590167674, −8.014387770357661, −7.675609784095514, −6.986799592825763, −5.967856200808485, −5.639918473637291, −5.005621975502211, −4.378836260470943, −3.575872561069478, −3.238024847148661, −1.914894487902528, −1.654132324160689, −0.5959619833953735, 0.5959619833953735, 1.654132324160689, 1.914894487902528, 3.238024847148661, 3.575872561069478, 4.378836260470943, 5.005621975502211, 5.639918473637291, 5.967856200808485, 6.986799592825763, 7.675609784095514, 8.014387770357661, 8.356016590167674, 8.978547120924256, 9.858604396618121, 10.42470824940304, 10.79606754031375, 11.41518617392460, 11.85383203732188, 12.34736839437848, 12.94697228382210, 13.74085118006921, 13.99892149620467, 14.59703381938023, 15.00287600118620

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