Properties

Label 2-2320-1.1-c1-0-44
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  + 1.30·3-s − 5-s − 4.30·7-s − 1.30·9-s + 4.60·11-s + 2.69·13-s − 1.30·15-s + 6.90·17-s − 6.60·19-s − 5.60·21-s − 5.30·23-s + 25-s − 5.60·27-s + 29-s − 2.90·31-s + 6·33-s + 4.30·35-s − 11.8·37-s + 3.51·39-s − 1.39·41-s + 0.302·43-s + 1.30·45-s + 11.5·49-s + 9·51-s + 6.90·53-s − 4.60·55-s − 8.60·57-s + ⋯
L(s)  = 1  + 0.752·3-s − 0.447·5-s − 1.62·7-s − 0.434·9-s + 1.38·11-s + 0.748·13-s − 0.336·15-s + 1.67·17-s − 1.51·19-s − 1.22·21-s − 1.10·23-s + 0.200·25-s − 1.07·27-s + 0.185·29-s − 0.522·31-s + 1.04·33-s + 0.727·35-s − 1.94·37-s + 0.562·39-s − 0.217·41-s + 0.0461·43-s + 0.194·45-s + 1.64·49-s + 1.26·51-s + 0.948·53-s − 0.621·55-s − 1.13·57-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 \( 1 - 1.30T + 3T^{2} \)
7 \( 1 + 4.30T + 7T^{2} \)
11 \( 1 - 4.60T + 11T^{2} \)
13 \( 1 - 2.69T + 13T^{2} \)
17 \( 1 - 6.90T + 17T^{2} \)
19 \( 1 + 6.60T + 19T^{2} \)
23 \( 1 + 5.30T + 23T^{2} \)
31 \( 1 + 2.90T + 31T^{2} \)
37 \( 1 + 11.8T + 37T^{2} \)
41 \( 1 + 1.39T + 41T^{2} \)
43 \( 1 - 0.302T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 6.90T + 53T^{2} \)
59 \( 1 + 9.90T + 59T^{2} \)
61 \( 1 + 13.9T + 61T^{2} \)
67 \( 1 + 5.21T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 15.5T + 73T^{2} \)
79 \( 1 + 5.90T + 79T^{2} \)
83 \( 1 - 1.39T + 83T^{2} \)
89 \( 1 + 7.39T + 89T^{2} \)
97 \( 1 - 11.9T + 97T^{2} \)
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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

−8.834622851480280446373823170001, −7.958019104008914974701589041704, −7.05945757398003687331649153180, −6.24309332866134048595864650211, −5.78304812045660763726211616487, −4.15562766505746859694673542676, −3.55575125325445793703370568540, −3.04437360152950489281911760891, −1.64817997764014666418367683924, 0, 1.64817997764014666418367683924, 3.04437360152950489281911760891, 3.55575125325445793703370568540, 4.15562766505746859694673542676, 5.78304812045660763726211616487, 6.24309332866134048595864650211, 7.05945757398003687331649153180, 7.958019104008914974701589041704, 8.834622851480280446373823170001

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