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Properties

Label 2-115920-1.1-c1-0-55
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 − 2·11-s − 6·13-s − 6·17-s − 23-s + 25-s − 4·29-s + 2·31-s − 35-s + 4·37-s − 2·41-s + 4·43-s + 49-s + 6·53-s − 2·55-s + 4·59-s − 6·65-s − 12·67-s − 8·71-s − 6·73-s + 2·77-s + 8·79-s + 10·83-s − 6·85-s + 2·89-s + 6·91-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s − 0.603·11-s − 1.66·13-s − 1.45·17-s − 0.208·23-s + 1/5·25-s − 0.742·29-s + 0.359·31-s − 0.169·35-s + 0.657·37-s − 0.312·41-s + 0.609·43-s + 1/7·49-s + 0.824·53-s − 0.269·55-s + 0.520·59-s − 0.744·65-s − 1.46·67-s − 0.949·71-s − 0.702·73-s + 0.227·77-s + 0.900·79-s + 1.09·83-s − 0.650·85-s + 0.211·89-s + 0.628·91-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 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 10 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
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

−13.71012875751685, −13.28125670134176, −13.01204273223997, −12.43068684825687, −11.93765302724262, −11.47964112154949, −10.84685730571013, −10.33794028388994, −10.02212230508025, −9.385752075077048, −9.080851272481762, −8.503043202041453, −7.791353832989670, −7.348618267122784, −6.940300861345402, −6.293966640487177, −5.807346714405570, −5.232287300296375, −4.610471374462934, −4.322837683224926, −3.418438519047745, −2.742581418811779, −2.292610019722604, −1.843709700620888, −0.6822775288877252, 0, 0.6822775288877252, 1.843709700620888, 2.292610019722604, 2.742581418811779, 3.418438519047745, 4.322837683224926, 4.610471374462934, 5.232287300296375, 5.807346714405570, 6.293966640487177, 6.940300861345402, 7.348618267122784, 7.791353832989670, 8.503043202041453, 9.080851272481762, 9.385752075077048, 10.02212230508025, 10.33794028388994, 10.84685730571013, 11.47964112154949, 11.93765302724262, 12.43068684825687, 13.01204273223997, 13.28125670134176, 13.71012875751685

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