Properties

Label 2-5415-1.1-c1-0-189
Degree $2$
Conductor $5415$
Sign $-1$
Analytic cond. $43.2389$
Root an. cond. $6.57563$
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 + 2·7-s + 9-s − 3·11-s − 2·12-s − 4·13-s + 15-s + 4·16-s − 2·20-s + 2·21-s − 6·23-s + 25-s + 27-s − 4·28-s + 3·29-s + 5·31-s − 3·33-s + 2·35-s − 2·36-s + 8·37-s − 4·39-s − 6·41-s − 4·43-s + 6·44-s + 45-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s + 0.447·5-s + 0.755·7-s + 1/3·9-s − 0.904·11-s − 0.577·12-s − 1.10·13-s + 0.258·15-s + 16-s − 0.447·20-s + 0.436·21-s − 1.25·23-s + 1/5·25-s + 0.192·27-s − 0.755·28-s + 0.557·29-s + 0.898·31-s − 0.522·33-s + 0.338·35-s − 1/3·36-s + 1.31·37-s − 0.640·39-s − 0.937·41-s − 0.609·43-s + 0.904·44-s + 0.149·45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5415\)    =    \(3 \cdot 5 \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(43.2389\)
Root analytic conductor: \(6.57563\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5415,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
19 \( 1 \)
good2 \( 1 + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 7 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−7.962785099581094910962508024465, −7.42587586337769953996503082539, −6.28336251268546415674272948170, −5.48884306612254360203795157278, −4.70016822763314478075826583630, −4.39794364107610280770594342431, −3.16663743578856928395398410129, −2.42006389342204391408519658880, −1.42284512898109478049476103017, 0, 1.42284512898109478049476103017, 2.42006389342204391408519658880, 3.16663743578856928395398410129, 4.39794364107610280770594342431, 4.70016822763314478075826583630, 5.48884306612254360203795157278, 6.28336251268546415674272948170, 7.42587586337769953996503082539, 7.962785099581094910962508024465

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