Properties

Label 2-2320-1.1-c1-0-50
Degree $2$
Conductor $2320$
Sign $-1$
Analytic cond. $18.5252$
Root an. cond. $4.30410$
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  + 0.806·3-s − 5-s + 4.15·7-s − 2.35·9-s − 2.80·11-s + 1.35·13-s − 0.806·15-s − 7.11·17-s − 3.76·19-s + 3.35·21-s − 4.80·23-s + 25-s − 4.31·27-s + 29-s − 0.231·31-s − 2.26·33-s − 4.15·35-s − 5.50·37-s + 1.08·39-s − 6.96·41-s + 3.19·43-s + 2.35·45-s − 6.41·47-s + 10.2·49-s − 5.73·51-s + 6.96·53-s + 2.80·55-s + ⋯
L(s)  = 1  + 0.465·3-s − 0.447·5-s + 1.57·7-s − 0.783·9-s − 0.846·11-s + 0.374·13-s − 0.208·15-s − 1.72·17-s − 0.864·19-s + 0.731·21-s − 1.00·23-s + 0.200·25-s − 0.829·27-s + 0.185·29-s − 0.0415·31-s − 0.393·33-s − 0.702·35-s − 0.905·37-s + 0.174·39-s − 1.08·41-s + 0.487·43-s + 0.350·45-s − 0.936·47-s + 1.46·49-s − 0.803·51-s + 0.956·53-s + 0.378·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2320\)    =    \(2^{4} \cdot 5 \cdot 29\)
Sign: $-1$
Analytic conductor: \(18.5252\)
Root analytic conductor: \(4.30410\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2320,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 + T \)
29 \( 1 - T \)
good3 \( 1 - 0.806T + 3T^{2} \)
7 \( 1 - 4.15T + 7T^{2} \)
11 \( 1 + 2.80T + 11T^{2} \)
13 \( 1 - 1.35T + 13T^{2} \)
17 \( 1 + 7.11T + 17T^{2} \)
19 \( 1 + 3.76T + 19T^{2} \)
23 \( 1 + 4.80T + 23T^{2} \)
31 \( 1 + 0.231T + 31T^{2} \)
37 \( 1 + 5.50T + 37T^{2} \)
41 \( 1 + 6.96T + 41T^{2} \)
43 \( 1 - 3.19T + 43T^{2} \)
47 \( 1 + 6.41T + 47T^{2} \)
53 \( 1 - 6.96T + 53T^{2} \)
59 \( 1 - 2.57T + 59T^{2} \)
61 \( 1 - 5.35T + 61T^{2} \)
67 \( 1 - 3.19T + 67T^{2} \)
71 \( 1 + 11.3T + 71T^{2} \)
73 \( 1 + 11.2T + 73T^{2} \)
79 \( 1 - 4.73T + 79T^{2} \)
83 \( 1 + 2.54T + 83T^{2} \)
89 \( 1 - 14.3T + 89T^{2} \)
97 \( 1 + 1.53T + 97T^{2} \)
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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

−8.456793229266646602142039033258, −8.135819676395002133051709048490, −7.27152483053227450223493339523, −6.27837355755144583870355559572, −5.29522980375089032546163196992, −4.58566172672220683836171139399, −3.78053824157967615217411963831, −2.52079415511911958956344111553, −1.84271647601281067439300861706, 0, 1.84271647601281067439300861706, 2.52079415511911958956344111553, 3.78053824157967615217411963831, 4.58566172672220683836171139399, 5.29522980375089032546163196992, 6.27837355755144583870355559572, 7.27152483053227450223493339523, 8.135819676395002133051709048490, 8.456793229266646602142039033258

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