Properties

Label 2-5415-1.1-c1-0-225
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 $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s − 4-s − 5-s − 6-s − 2·7-s − 3·8-s + 9-s − 10-s − 6·11-s + 12-s − 2·14-s + 15-s − 16-s − 6·17-s + 18-s + 20-s + 2·21-s − 6·22-s − 8·23-s + 3·24-s + 25-s − 27-s + 2·28-s − 4·29-s + 30-s + 5·32-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.755·7-s − 1.06·8-s + 1/3·9-s − 0.316·10-s − 1.80·11-s + 0.288·12-s − 0.534·14-s + 0.258·15-s − 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.223·20-s + 0.436·21-s − 1.27·22-s − 1.66·23-s + 0.612·24-s + 1/5·25-s − 0.192·27-s + 0.377·28-s − 0.742·29-s + 0.182·30-s + 0.883·32-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: $\chi_{5415} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
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 \( 1 - T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−7.42195896981757685932141018119, −6.45754305631640026627862079587, −5.96416216153721976810548337042, −5.13514216792987632614840809373, −4.62770227716381985665512695248, −3.81467929255751360490045430008, −3.05985687846521158771439909949, −2.09668804479314175714108769749, 0, 0, 2.09668804479314175714108769749, 3.05985687846521158771439909949, 3.81467929255751360490045430008, 4.62770227716381985665512695248, 5.13514216792987632614840809373, 5.96416216153721976810548337042, 6.45754305631640026627862079587, 7.42195896981757685932141018119

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