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Properties

Label 2-115920-1.1-c1-0-80
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 − 6·17-s − 4·19-s − 23-s + 25-s − 6·29-s − 4·31-s − 35-s + 2·37-s − 10·41-s + 8·43-s + 4·47-s + 49-s + 6·53-s − 4·59-s + 2·61-s + 2·65-s + 8·67-s − 8·71-s + 10·73-s + 12·83-s − 6·85-s + 2·89-s − 2·91-s − 4·95-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s + 0.554·13-s − 1.45·17-s − 0.917·19-s − 0.208·23-s + 1/5·25-s − 1.11·29-s − 0.718·31-s − 0.169·35-s + 0.328·37-s − 1.56·41-s + 1.21·43-s + 0.583·47-s + 1/7·49-s + 0.824·53-s − 0.520·59-s + 0.256·61-s + 0.248·65-s + 0.977·67-s − 0.949·71-s + 1.17·73-s + 1.31·83-s − 0.650·85-s + 0.211·89-s − 0.209·91-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 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
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

−13.70654072127094, −13.24586719181769, −13.10733563818146, −12.41980383203023, −12.00498352409127, −11.26296112956477, −10.82501945489519, −10.66621379903344, −9.828902651779646, −9.474452189310084, −8.824731494159938, −8.659271498939743, −7.949857388310082, −7.261537866015025, −6.829853766552168, −6.292591218989643, −5.888035073178335, −5.296030111160467, −4.653254778780688, −4.009643827609064, −3.654335693635883, −2.829373964184618, −2.107371768041765, −1.850032547063631, −0.7834739696059799, 0, 0.7834739696059799, 1.850032547063631, 2.107371768041765, 2.829373964184618, 3.654335693635883, 4.009643827609064, 4.653254778780688, 5.296030111160467, 5.888035073178335, 6.292591218989643, 6.829853766552168, 7.261537866015025, 7.949857388310082, 8.659271498939743, 8.824731494159938, 9.474452189310084, 9.828902651779646, 10.66621379903344, 10.82501945489519, 11.26296112956477, 12.00498352409127, 12.41980383203023, 13.10733563818146, 13.24586719181769, 13.70654072127094

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