Properties

Label 2-30960-1.1-c1-0-54
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 $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 4·7-s − 2·11-s + 2·13-s − 6·19-s + 25-s − 10·29-s − 8·31-s + 4·35-s − 4·37-s + 10·41-s + 43-s + 9·49-s − 12·53-s + 2·55-s − 6·59-s − 10·61-s − 2·65-s − 12·67-s + 4·71-s − 8·73-s + 8·77-s + 16·79-s − 12·83-s + 10·89-s − 8·91-s + 6·95-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.51·7-s − 0.603·11-s + 0.554·13-s − 1.37·19-s + 1/5·25-s − 1.85·29-s − 1.43·31-s + 0.676·35-s − 0.657·37-s + 1.56·41-s + 0.152·43-s + 9/7·49-s − 1.64·53-s + 0.269·55-s − 0.781·59-s − 1.28·61-s − 0.248·65-s − 1.46·67-s + 0.474·71-s − 0.936·73-s + 0.911·77-s + 1.80·79-s − 1.31·83-s + 1.05·89-s − 0.838·91-s + 0.615·95-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: \(2\)
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 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−15.59838823999830, −15.22267051427458, −14.66664611630287, −14.03117355465891, −13.25588149381954, −12.97339281198927, −12.61538006320702, −12.08605588287216, −11.16023651168422, −10.82510260695572, −10.43001440388356, −9.572704368790627, −9.145002654369899, −8.790841183608106, −7.761178336262213, −7.577426118901042, −6.766862192879689, −6.169783670735022, −5.821240756443732, −4.990704951397567, −4.138725495295343, −3.673764055084741, −3.093146114608665, −2.342290056505818, −1.483029068902879, 0, 0, 1.483029068902879, 2.342290056505818, 3.093146114608665, 3.673764055084741, 4.138725495295343, 4.990704951397567, 5.821240756443732, 6.169783670735022, 6.766862192879689, 7.577426118901042, 7.761178336262213, 8.790841183608106, 9.145002654369899, 9.572704368790627, 10.43001440388356, 10.82510260695572, 11.16023651168422, 12.08605588287216, 12.61538006320702, 12.97339281198927, 13.25588149381954, 14.03117355465891, 14.66664611630287, 15.22267051427458, 15.59838823999830

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