Properties

Label 2-115920-1.1-c1-0-110
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 + 4·17-s − 8·19-s + 23-s + 25-s + 4·31-s − 35-s + 6·37-s + 4·41-s + 4·43-s + 12·47-s + 49-s + 4·53-s − 2·61-s − 2·65-s − 12·67-s − 8·71-s + 16·73-s + 4·79-s − 4·85-s − 6·89-s + 2·91-s + 8·95-s + 2·97-s + 101-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.377·7-s + 0.554·13-s + 0.970·17-s − 1.83·19-s + 0.208·23-s + 1/5·25-s + 0.718·31-s − 0.169·35-s + 0.986·37-s + 0.624·41-s + 0.609·43-s + 1.75·47-s + 1/7·49-s + 0.549·53-s − 0.256·61-s − 0.248·65-s − 1.46·67-s − 0.949·71-s + 1.87·73-s + 0.450·79-s − 0.433·85-s − 0.635·89-s + 0.209·91-s + 0.820·95-s + 0.203·97-s + 0.0995·101-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 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
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

−13.84136586996900, −13.36268318984149, −12.83839280666805, −12.27554045072579, −12.07391869560846, −11.34007649524851, −10.91533252467585, −10.52885445747391, −10.09828265489296, −9.261787834442401, −8.969621711534227, −8.351413562992763, −7.915019392563496, −7.539876290762394, −6.842928711519449, −6.273685102339778, −5.870033157520541, −5.229068038676552, −4.578160911472734, −3.995141688599634, −3.789277944959229, −2.643089721855016, −2.533870155779404, −1.431482347256272, −0.9469237620189366, 0, 0.9469237620189366, 1.431482347256272, 2.533870155779404, 2.643089721855016, 3.789277944959229, 3.995141688599634, 4.578160911472734, 5.229068038676552, 5.870033157520541, 6.273685102339778, 6.842928711519449, 7.539876290762394, 7.915019392563496, 8.351413562992763, 8.969621711534227, 9.261787834442401, 10.09828265489296, 10.52885445747391, 10.91533252467585, 11.34007649524851, 12.07391869560846, 12.27554045072579, 12.83839280666805, 13.36268318984149, 13.84136586996900

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