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Properties

Label 2-388815-1.1-c1-0-36
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 + 6·11-s + 2·12-s − 4·13-s + 15-s + 4·16-s − 3·17-s − 6·19-s + 2·20-s + 25-s − 27-s + 2·29-s − 8·31-s − 6·33-s − 2·36-s − 6·37-s + 4·39-s + 9·41-s + 9·43-s − 12·44-s − 45-s − 8·47-s − 4·48-s + 3·51-s + ⋯
L(s)  = 1  − 0.577·3-s − 4-s − 0.447·5-s + 1/3·9-s + 1.80·11-s + 0.577·12-s − 1.10·13-s + 0.258·15-s + 16-s − 0.727·17-s − 1.37·19-s + 0.447·20-s + 1/5·25-s − 0.192·27-s + 0.371·29-s − 1.43·31-s − 1.04·33-s − 1/3·36-s − 0.986·37-s + 0.640·39-s + 1.40·41-s + 1.37·43-s − 1.80·44-s − 0.149·45-s − 1.16·47-s − 0.577·48-s + 0.420·51-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 \( 1 - 6 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 - 9 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 11 T + p T^{2} \)
59 \( 1 - 5 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + T + p T^{2} \)
71 \( 1 - T + p T^{2} \)
73 \( 1 + 12 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 19 T + p T^{2} \)
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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

−12.63065091974251, −12.21405546266229, −11.96355084860876, −11.30043839826388, −10.91014784045782, −10.50089839183237, −9.938108294657298, −9.403451502131360, −9.024821251943848, −8.834514065963272, −8.189892486343257, −7.623513684516776, −7.087821866130400, −6.768716836088893, −6.182160125692615, −5.725482070295136, −5.179978032274375, −4.523964915135071, −4.270425004019348, −3.959015454747411, −3.358705302729857, −2.559027747613923, −1.882728047781016, −1.297196539694008, −0.5486499280118065, 0, 0.5486499280118065, 1.297196539694008, 1.882728047781016, 2.559027747613923, 3.358705302729857, 3.959015454747411, 4.270425004019348, 4.523964915135071, 5.179978032274375, 5.725482070295136, 6.182160125692615, 6.768716836088893, 7.087821866130400, 7.623513684516776, 8.189892486343257, 8.834514065963272, 9.024821251943848, 9.403451502131360, 9.938108294657298, 10.50089839183237, 10.91014784045782, 11.30043839826388, 11.96355084860876, 12.21405546266229, 12.63065091974251

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