Properties

Label 2-30960-1.1-c1-0-41
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 2·7-s + 11-s − 5·13-s + 3·17-s + 6·19-s + 5·23-s + 25-s + 7·31-s − 2·35-s − 8·37-s − 5·41-s − 43-s − 8·47-s − 3·49-s + 9·53-s − 55-s − 4·59-s + 8·61-s + 5·65-s − 11·67-s − 14·71-s − 4·73-s + 2·77-s − 16·79-s + 83-s − 3·85-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.755·7-s + 0.301·11-s − 1.38·13-s + 0.727·17-s + 1.37·19-s + 1.04·23-s + 1/5·25-s + 1.25·31-s − 0.338·35-s − 1.31·37-s − 0.780·41-s − 0.152·43-s − 1.16·47-s − 3/7·49-s + 1.23·53-s − 0.134·55-s − 0.520·59-s + 1.02·61-s + 0.620·65-s − 1.34·67-s − 1.66·71-s − 0.468·73-s + 0.227·77-s − 1.80·79-s + 0.109·83-s − 0.325·85-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: \(1\)
Selberg data: \((2,\ 30960,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 + T \)
43 \( 1 + T \)
good7 \( 1 - 2 T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 5 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 11 T + p T^{2} \)
71 \( 1 + 14 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 17 T + p T^{2} \)
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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

−15.19675006388639, −14.76743383596251, −14.50482109484219, −13.77580816841960, −13.38415399358001, −12.54964337393122, −12.05155024279230, −11.68171786213508, −11.34953397840342, −10.36773908411022, −10.11349746691019, −9.454106546963175, −8.836863638264259, −8.189861930993961, −7.758987598774623, −6.985541297305498, −6.926383112133144, −5.696938501146570, −5.229453826722494, −4.736887575573722, −4.128427034159326, −3.094075646602868, −2.898860791717688, −1.695381175725021, −1.120713568167655, 0, 1.120713568167655, 1.695381175725021, 2.898860791717688, 3.094075646602868, 4.128427034159326, 4.736887575573722, 5.229453826722494, 5.696938501146570, 6.926383112133144, 6.985541297305498, 7.758987598774623, 8.189861930993961, 8.836863638264259, 9.454106546963175, 10.11349746691019, 10.36773908411022, 11.34953397840342, 11.68171786213508, 12.05155024279230, 12.54964337393122, 13.38415399358001, 13.77580816841960, 14.50482109484219, 14.76743383596251, 15.19675006388639

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