Properties

Label 2-188760-1.1-c1-0-12
Degree $2$
Conductor $188760$
Sign $1$
Analytic cond. $1507.25$
Root an. cond. $38.8233$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s + 2·7-s + 9-s − 13-s + 15-s − 3·17-s + 4·19-s + 2·21-s − 9·23-s + 25-s + 27-s − 8·29-s − 31-s + 2·35-s + 6·37-s − 39-s − 8·41-s + 12·43-s + 45-s − 13·47-s − 3·49-s − 3·51-s + 53-s + 4·57-s + 6·59-s + 5·61-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s − 0.277·13-s + 0.258·15-s − 0.727·17-s + 0.917·19-s + 0.436·21-s − 1.87·23-s + 1/5·25-s + 0.192·27-s − 1.48·29-s − 0.179·31-s + 0.338·35-s + 0.986·37-s − 0.160·39-s − 1.24·41-s + 1.82·43-s + 0.149·45-s − 1.89·47-s − 3/7·49-s − 0.420·51-s + 0.137·53-s + 0.529·57-s + 0.781·59-s + 0.640·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(188760\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 11^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(1507.25\)
Root analytic conductor: \(38.8233\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{188760} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 188760,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.941425614\)
\(L(\frac12)\) \(\approx\) \(2.941425614\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
11 \( 1 \)
13 \( 1 + T \)
good7 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 13 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−13.18305855652320, −12.81247265844279, −12.11222142522332, −11.72596016502481, −11.17738360141492, −10.90975093005720, −10.06040856176796, −9.754863173806502, −9.494095269707853, −8.712017563039784, −8.421339095974504, −7.832036524552219, −7.458720781169794, −6.981858615907669, −6.251015813567192, −5.818765605529589, −5.284756691633901, −4.704367276550955, −4.190221233776710, −3.672792096898202, −3.026458730998432, −2.322488923115575, −1.889908528648170, −1.436941721391191, −0.4426069685978443, 0.4426069685978443, 1.436941721391191, 1.889908528648170, 2.322488923115575, 3.026458730998432, 3.672792096898202, 4.190221233776710, 4.704367276550955, 5.284756691633901, 5.818765605529589, 6.251015813567192, 6.981858615907669, 7.458720781169794, 7.832036524552219, 8.421339095974504, 8.712017563039784, 9.494095269707853, 9.754863173806502, 10.06040856176796, 10.90975093005720, 11.17738360141492, 11.72596016502481, 12.11222142522332, 12.81247265844279, 13.18305855652320

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