Properties

Label 2-30960-1.1-c1-0-33
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 − 6·13-s + 2·17-s − 2·19-s + 6·23-s + 25-s + 6·29-s − 4·31-s − 8·37-s + 8·41-s − 43-s + 6·47-s − 7·49-s + 6·53-s + 4·59-s − 14·61-s − 6·65-s + 4·67-s − 8·71-s − 4·73-s + 12·79-s − 2·83-s + 2·85-s − 14·89-s − 2·95-s − 2·97-s + 101-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.66·13-s + 0.485·17-s − 0.458·19-s + 1.25·23-s + 1/5·25-s + 1.11·29-s − 0.718·31-s − 1.31·37-s + 1.24·41-s − 0.152·43-s + 0.875·47-s − 49-s + 0.824·53-s + 0.520·59-s − 1.79·61-s − 0.744·65-s + 0.488·67-s − 0.949·71-s − 0.468·73-s + 1.35·79-s − 0.219·83-s + 0.216·85-s − 1.48·89-s − 0.205·95-s − 0.203·97-s + 0.0995·101-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 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−15.28378229801124, −14.75773340735733, −14.33344244441749, −13.92180632614447, −13.13770143960549, −12.74074367449530, −12.17537127248431, −11.82899697405605, −10.95865955493865, −10.48103448632707, −10.08419931990614, −9.268783800193478, −9.128841210404676, −8.268744544656148, −7.659808526170077, −7.051140205299876, −6.689382473364464, −5.810855004223408, −5.271810618740199, −4.762339779178217, −4.119550862914375, −3.134488683539288, −2.663115368826758, −1.927120399106757, −1.043679544610964, 0, 1.043679544610964, 1.927120399106757, 2.663115368826758, 3.134488683539288, 4.119550862914375, 4.762339779178217, 5.271810618740199, 5.810855004223408, 6.689382473364464, 7.051140205299876, 7.659808526170077, 8.268744544656148, 9.128841210404676, 9.268783800193478, 10.08419931990614, 10.48103448632707, 10.95865955493865, 11.82899697405605, 12.17537127248431, 12.74074367449530, 13.13770143960549, 13.92180632614447, 14.33344244441749, 14.75773340735733, 15.28378229801124

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