Properties

Label 2-2366-1.1-c1-0-74
Degree $2$
Conductor $2366$
Sign $-1$
Analytic cond. $18.8926$
Root an. cond. $4.34656$
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  + 2-s − 0.198·3-s + 4-s + 0.890·5-s − 0.198·6-s − 7-s + 8-s − 2.96·9-s + 0.890·10-s + 0.664·11-s − 0.198·12-s − 14-s − 0.176·15-s + 16-s − 2.44·17-s − 2.96·18-s − 8.63·19-s + 0.890·20-s + 0.198·21-s + 0.664·22-s − 7.60·23-s − 0.198·24-s − 4.20·25-s + 1.18·27-s − 28-s + 3.60·29-s − 0.176·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.114·3-s + 0.5·4-s + 0.398·5-s − 0.0808·6-s − 0.377·7-s + 0.353·8-s − 0.986·9-s + 0.281·10-s + 0.200·11-s − 0.0571·12-s − 0.267·14-s − 0.0455·15-s + 0.250·16-s − 0.593·17-s − 0.697·18-s − 1.98·19-s + 0.199·20-s + 0.0432·21-s + 0.141·22-s − 1.58·23-s − 0.0404·24-s − 0.841·25-s + 0.227·27-s − 0.188·28-s + 0.669·29-s − 0.0321·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2366\)    =    \(2 \cdot 7 \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(18.8926\)
Root analytic conductor: \(4.34656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2366,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 - T \)
7 \( 1 + T \)
13 \( 1 \)
good3 \( 1 + 0.198T + 3T^{2} \)
5 \( 1 - 0.890T + 5T^{2} \)
11 \( 1 - 0.664T + 11T^{2} \)
17 \( 1 + 2.44T + 17T^{2} \)
19 \( 1 + 8.63T + 19T^{2} \)
23 \( 1 + 7.60T + 23T^{2} \)
29 \( 1 - 3.60T + 29T^{2} \)
31 \( 1 + 1.50T + 31T^{2} \)
37 \( 1 + 5.60T + 37T^{2} \)
41 \( 1 - 7.83T + 41T^{2} \)
43 \( 1 - 7.46T + 43T^{2} \)
47 \( 1 + 1.20T + 47T^{2} \)
53 \( 1 + 4.89T + 53T^{2} \)
59 \( 1 + 12.8T + 59T^{2} \)
61 \( 1 - 7.70T + 61T^{2} \)
67 \( 1 - 6.07T + 67T^{2} \)
71 \( 1 + 6.27T + 71T^{2} \)
73 \( 1 - 3.67T + 73T^{2} \)
79 \( 1 - 4.37T + 79T^{2} \)
83 \( 1 + 2.92T + 83T^{2} \)
89 \( 1 + 12.5T + 89T^{2} \)
97 \( 1 + 10.4T + 97T^{2} \)
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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

−8.553002730704364656247266730966, −7.85364217211133381955964297298, −6.68191337384785991318303201294, −6.14880098932720461896367868909, −5.63310137788663928691385174312, −4.47472768324780206959237674319, −3.83907833483421840763890757127, −2.65406921201735599756584591509, −1.95233551606990059219029172941, 0, 1.95233551606990059219029172941, 2.65406921201735599756584591509, 3.83907833483421840763890757127, 4.47472768324780206959237674319, 5.63310137788663928691385174312, 6.14880098932720461896367868909, 6.68191337384785991318303201294, 7.85364217211133381955964297298, 8.553002730704364656247266730966

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