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Properties

Label 2-388815-1.1-c1-0-35
Degree $2$
Conductor $388815$
Sign $-1$
Analytic cond. $3104.70$
Root an. cond. $55.7198$
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 − 2·4-s + 5-s + 9-s − 4·11-s − 2·12-s − 5·13-s + 15-s + 4·16-s − 7·17-s − 2·20-s + 25-s + 27-s − 5·29-s + 2·31-s − 4·33-s − 2·36-s − 2·37-s − 5·39-s + 4·41-s + 10·43-s + 8·44-s + 45-s − 3·47-s + 4·48-s − 7·51-s + 10·52-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s + 0.447·5-s + 1/3·9-s − 1.20·11-s − 0.577·12-s − 1.38·13-s + 0.258·15-s + 16-s − 1.69·17-s − 0.447·20-s + 1/5·25-s + 0.192·27-s − 0.928·29-s + 0.359·31-s − 0.696·33-s − 1/3·36-s − 0.328·37-s − 0.800·39-s + 0.624·41-s + 1.52·43-s + 1.20·44-s + 0.149·45-s − 0.437·47-s + 0.577·48-s − 0.980·51-s + 1.38·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 388815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(388815\)    =    \(3 \cdot 5 \cdot 7^{2} \cdot 23^{2}\)
Sign: $-1$
Analytic conductor: \(3104.70\)
Root analytic conductor: \(55.7198\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 388815,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 + T + p T^{2} \)
79 \( 1 + 9 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
97 \( 1 + 3 T + p T^{2} \)
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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

−12.86648951637940, −12.51108446472255, −11.85309228840926, −11.25585137330006, −10.72621492134094, −10.37598658600873, −9.761496877975282, −9.613691090483334, −9.051082865402075, −8.656204123399252, −8.250212962039710, −7.654782845653010, −7.311312941809330, −6.862643301566962, −6.118504887050991, −5.585546307948587, −5.171970901298222, −4.703810846867527, −4.232688152443198, −3.865619210489166, −2.894005884087454, −2.668577389040848, −2.148727530492091, −1.509146590732447, −0.5476050664412494, 0, 0.5476050664412494, 1.509146590732447, 2.148727530492091, 2.668577389040848, 2.894005884087454, 3.865619210489166, 4.232688152443198, 4.703810846867527, 5.171970901298222, 5.585546307948587, 6.118504887050991, 6.862643301566962, 7.311312941809330, 7.654782845653010, 8.250212962039710, 8.656204123399252, 9.051082865402075, 9.613691090483334, 9.761496877975282, 10.37598658600873, 10.72621492134094, 11.25585137330006, 11.85309228840926, 12.51108446472255, 12.86648951637940

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