Properties

Label 2-5415-1.1-c1-0-195
Degree $2$
Conductor $5415$
Sign $-1$
Analytic cond. $43.2389$
Root an. cond. $6.57563$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73·2-s − 3-s + 0.999·4-s + 5-s − 1.73·6-s − 2.73·7-s − 1.73·8-s + 9-s + 1.73·10-s + 4.73·11-s − 0.999·12-s − 0.732·13-s − 4.73·14-s − 15-s − 5·16-s + 1.73·18-s + 0.999·20-s + 2.73·21-s + 8.19·22-s + 3.46·23-s + 1.73·24-s + 25-s − 1.26·26-s − 27-s − 2.73·28-s − 8.19·29-s − 1.73·30-s + ⋯
L(s)  = 1  + 1.22·2-s − 0.577·3-s + 0.499·4-s + 0.447·5-s − 0.707·6-s − 1.03·7-s − 0.612·8-s + 0.333·9-s + 0.547·10-s + 1.42·11-s − 0.288·12-s − 0.203·13-s − 1.26·14-s − 0.258·15-s − 1.25·16-s + 0.408·18-s + 0.223·20-s + 0.596·21-s + 1.74·22-s + 0.722·23-s + 0.353·24-s + 0.200·25-s − 0.248·26-s − 0.192·27-s − 0.516·28-s − 1.52·29-s − 0.316·30-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: no
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 \( 1 - 1.73T + 2T^{2} \)
7 \( 1 + 2.73T + 7T^{2} \)
11 \( 1 - 4.73T + 11T^{2} \)
13 \( 1 + 0.732T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
23 \( 1 - 3.46T + 23T^{2} \)
29 \( 1 + 8.19T + 29T^{2} \)
31 \( 1 + 8.92T + 31T^{2} \)
37 \( 1 - 6.19T + 37T^{2} \)
41 \( 1 + 1.26T + 41T^{2} \)
43 \( 1 - 4.19T + 43T^{2} \)
47 \( 1 + 3.46T + 47T^{2} \)
53 \( 1 - 9.46T + 53T^{2} \)
59 \( 1 + 2.53T + 59T^{2} \)
61 \( 1 + 6.53T + 61T^{2} \)
67 \( 1 + 8T + 67T^{2} \)
71 \( 1 - 4.39T + 71T^{2} \)
73 \( 1 + 16.9T + 73T^{2} \)
79 \( 1 - 10.9T + 79T^{2} \)
83 \( 1 + 12.9T + 83T^{2} \)
89 \( 1 + 10.7T + 89T^{2} \)
97 \( 1 - 6.19T + 97T^{2} \)
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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

−7.32478401844020461457781769827, −6.86499150379127540525290213246, −6.04058574568757094916242401717, −5.79693992146126944657838424507, −4.91799957204996053220937668842, −4.05751383096896134034704258424, −3.56684986462496203530113572455, −2.65796349454318247575124955266, −1.46539775694348188894541597264, 0, 1.46539775694348188894541597264, 2.65796349454318247575124955266, 3.56684986462496203530113572455, 4.05751383096896134034704258424, 4.91799957204996053220937668842, 5.79693992146126944657838424507, 6.04058574568757094916242401717, 6.86499150379127540525290213246, 7.32478401844020461457781769827

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