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Properties

Label 2-115920-1.1-c1-0-89
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

−14.07312915727457, −13.25419910007234, −12.79578169889173, −12.25789491303711, −11.96348565100890, −11.69452852682687, −10.79241602424681, −10.49807681534064, −9.818419680458543, −9.475638053293833, −9.147193374238898, −8.162336879756061, −8.032405097083206, −7.242237791431523, −7.089241753782975, −6.297677131141576, −5.834796769530814, −5.173237920179571, −4.632845447266960, −4.180113714805718, −3.434530685708180, −2.911933130726044, −2.447808298627522, −1.478645463632080, −0.8354855898467271, 0, 0.8354855898467271, 1.478645463632080, 2.447808298627522, 2.911933130726044, 3.434530685708180, 4.180113714805718, 4.632845447266960, 5.173237920179571, 5.834796769530814, 6.297677131141576, 7.089241753782975, 7.242237791431523, 8.032405097083206, 8.162336879756061, 9.147193374238898, 9.475638053293833, 9.818419680458543, 10.49807681534064, 10.79241602424681, 11.69452852682687, 11.96348565100890, 12.25789491303711, 12.79578169889173, 13.25419910007234, 14.07312915727457

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