Properties

Label 2-115920-1.1-c1-0-124
Degree $2$
Conductor $115920$
Sign $-1$
Analytic cond. $925.625$
Root an. cond. $30.4240$
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 + 7-s + 2·13-s + 2·17-s − 4·19-s + 23-s + 25-s + 2·29-s + 35-s + 2·37-s − 2·41-s − 4·43-s + 12·47-s + 49-s − 2·53-s − 12·59-s − 6·61-s + 2·65-s + 4·67-s − 10·73-s + 4·79-s − 12·83-s + 2·85-s − 6·89-s + 2·91-s − 4·95-s − 2·97-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s + 0.554·13-s + 0.485·17-s − 0.917·19-s + 0.208·23-s + 1/5·25-s + 0.371·29-s + 0.169·35-s + 0.328·37-s − 0.312·41-s − 0.609·43-s + 1.75·47-s + 1/7·49-s − 0.274·53-s − 1.56·59-s − 0.768·61-s + 0.248·65-s + 0.488·67-s − 1.17·73-s + 0.450·79-s − 1.31·83-s + 0.216·85-s − 0.635·89-s + 0.209·91-s − 0.410·95-s − 0.203·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115920\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 23\)
Sign: $-1$
Analytic conductor: \(925.625\)
Root analytic conductor: \(30.4240\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{115920} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 115920,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
7 \( 1 - T \)
23 \( 1 - T \)
good11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−13.80206607098479, −13.46437688647388, −12.87008577476734, −12.36256408468128, −12.04956920690305, −11.18654398448867, −11.09095251969459, −10.25119301075788, −10.20353923937497, −9.342081938130652, −8.933995195744062, −8.482083789616228, −7.938053321177969, −7.419390460437242, −6.815749141382831, −6.262980957506999, −5.817717266832816, −5.315231446914200, −4.577954361571670, −4.248209960639516, −3.459571965957366, −2.893705851122948, −2.231742208169894, −1.560807980261131, −1.006133631507916, 0, 1.006133631507916, 1.560807980261131, 2.231742208169894, 2.893705851122948, 3.459571965957366, 4.248209960639516, 4.577954361571670, 5.315231446914200, 5.817717266832816, 6.262980957506999, 6.815749141382831, 7.419390460437242, 7.938053321177969, 8.482083789616228, 8.933995195744062, 9.342081938130652, 10.20353923937497, 10.25119301075788, 11.09095251969459, 11.18654398448867, 12.04956920690305, 12.36256408468128, 12.87008577476734, 13.46437688647388, 13.80206607098479

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