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Properties

Label 2-115920-1.1-c1-0-75
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 − 6·13-s − 6·17-s + 4·19-s − 23-s + 25-s + 2·29-s − 4·31-s − 35-s + 2·37-s − 2·41-s − 4·47-s + 49-s + 14·53-s + 12·59-s + 2·61-s − 6·65-s − 16·67-s − 8·71-s + 10·73-s + 8·79-s + 4·83-s − 6·85-s − 6·89-s + 6·91-s + 4·95-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s − 1.66·13-s − 1.45·17-s + 0.917·19-s − 0.208·23-s + 1/5·25-s + 0.371·29-s − 0.718·31-s − 0.169·35-s + 0.328·37-s − 0.312·41-s − 0.583·47-s + 1/7·49-s + 1.92·53-s + 1.56·59-s + 0.256·61-s − 0.744·65-s − 1.95·67-s − 0.949·71-s + 1.17·73-s + 0.900·79-s + 0.439·83-s − 0.650·85-s − 0.635·89-s + 0.628·91-s + 0.410·95-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: Trivial
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 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
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

−13.73221553699157, −13.31745246122470, −13.01693777455503, −12.36200943784485, −11.88989530956387, −11.57162975787814, −10.86828607131375, −10.32804638390821, −9.934926349075995, −9.493787049107088, −8.974285999942711, −8.595732660238058, −7.751214392410647, −7.380570980491842, −6.800044674097192, −6.493793108604080, −5.659148637249898, −5.285801576119266, −4.712030142877624, −4.178027862772256, −3.485368622143924, −2.721493348417729, −2.355583316021611, −1.744132533971670, −0.7541857395637584, 0, 0.7541857395637584, 1.744132533971670, 2.355583316021611, 2.721493348417729, 3.485368622143924, 4.178027862772256, 4.712030142877624, 5.285801576119266, 5.659148637249898, 6.493793108604080, 6.800044674097192, 7.380570980491842, 7.751214392410647, 8.595732660238058, 8.974285999942711, 9.493787049107088, 9.934926349075995, 10.32804638390821, 10.86828607131375, 11.57162975787814, 11.88989530956387, 12.36200943784485, 13.01693777455503, 13.31745246122470, 13.73221553699157

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