Properties

Label 2-11760-1.1-c1-0-70
Degree $2$
Conductor $11760$
Sign $-1$
Analytic cond. $93.9040$
Root an. cond. $9.69041$
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  + 3-s + 5-s + 9-s − 2·13-s + 15-s + 6·17-s − 4·19-s + 25-s + 27-s − 6·29-s − 4·31-s + 2·37-s − 2·39-s − 6·41-s − 8·43-s + 45-s − 12·47-s + 6·51-s + 6·53-s − 4·57-s − 12·59-s − 2·61-s − 2·65-s − 8·67-s − 14·73-s + 75-s + 16·79-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.554·13-s + 0.258·15-s + 1.45·17-s − 0.917·19-s + 1/5·25-s + 0.192·27-s − 1.11·29-s − 0.718·31-s + 0.328·37-s − 0.320·39-s − 0.937·41-s − 1.21·43-s + 0.149·45-s − 1.75·47-s + 0.840·51-s + 0.824·53-s − 0.529·57-s − 1.56·59-s − 0.256·61-s − 0.248·65-s − 0.977·67-s − 1.63·73-s + 0.115·75-s + 1.80·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(11760\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(93.9040\)
Root analytic conductor: \(9.69041\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{11760} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 11760,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 \)
good11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−16.65923486397524, −16.36636522893473, −15.26951460723723, −14.88168406505266, −14.59780154542738, −13.83365536132892, −13.30589981083666, −12.79977899153455, −12.17278071016755, −11.62756652869789, −10.73660559095293, −10.28122212767041, −9.605093206856188, −9.248561191636635, −8.396791088790065, −7.907116623875908, −7.256162742972956, −6.586220244527530, −5.823254019156098, −5.182624051924949, −4.474606958703111, −3.539659067778729, −3.039946646525370, −2.035155177139918, −1.454593903281767, 0, 1.454593903281767, 2.035155177139918, 3.039946646525370, 3.539659067778729, 4.474606958703111, 5.182624051924949, 5.823254019156098, 6.586220244527530, 7.256162742972956, 7.907116623875908, 8.396791088790065, 9.248561191636635, 9.605093206856188, 10.28122212767041, 10.73660559095293, 11.62756652869789, 12.17278071016755, 12.79977899153455, 13.30589981083666, 13.83365536132892, 14.59780154542738, 14.88168406505266, 15.26951460723723, 16.36636522893473, 16.65923486397524

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