Properties

Label 2-2320-1.1-c1-0-22
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 5-s − 0.828·7-s + 9-s + 4.82·11-s − 2·13-s + 2·15-s − 2.82·17-s − 0.828·19-s − 1.65·21-s + 8.82·23-s + 25-s − 4·27-s + 29-s + 10.4·31-s + 9.65·33-s − 0.828·35-s + 8.48·37-s − 4·39-s − 6·41-s + 6·43-s + 45-s + 0.343·47-s − 6.31·49-s − 5.65·51-s + 7.65·53-s + 4.82·55-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.447·5-s − 0.313·7-s + 0.333·9-s + 1.45·11-s − 0.554·13-s + 0.516·15-s − 0.685·17-s − 0.190·19-s − 0.361·21-s + 1.84·23-s + 0.200·25-s − 0.769·27-s + 0.185·29-s + 1.88·31-s + 1.68·33-s − 0.140·35-s + 1.39·37-s − 0.640·39-s − 0.937·41-s + 0.914·43-s + 0.149·45-s + 0.0500·47-s − 0.901·49-s − 0.792·51-s + 1.05·53-s + 0.651·55-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: \(0\)
Selberg data: \((2,\ 2320,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.030917497\)
\(L(\frac12)\) \(\approx\) \(3.030917497\)
\(L(\frac{3}{2})\) 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 \( 1 - 2T + 3T^{2} \)
7 \( 1 + 0.828T + 7T^{2} \)
11 \( 1 - 4.82T + 11T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 + 2.82T + 17T^{2} \)
19 \( 1 + 0.828T + 19T^{2} \)
23 \( 1 - 8.82T + 23T^{2} \)
31 \( 1 - 10.4T + 31T^{2} \)
37 \( 1 - 8.48T + 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 6T + 43T^{2} \)
47 \( 1 - 0.343T + 47T^{2} \)
53 \( 1 - 7.65T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 7.65T + 61T^{2} \)
67 \( 1 - 10.4T + 67T^{2} \)
71 \( 1 + 7.31T + 71T^{2} \)
73 \( 1 + 8.48T + 73T^{2} \)
79 \( 1 + 14.4T + 79T^{2} \)
83 \( 1 + 12.8T + 83T^{2} \)
89 \( 1 - 3.65T + 89T^{2} \)
97 \( 1 - 4.48T + 97T^{2} \)
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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

−8.827440748797551310744489617320, −8.608338612079200641754060102542, −7.44408125712967954118004582988, −6.74168988426297517090799028191, −6.07102379371770908331786128296, −4.83969109724384096480112467617, −4.04578995504124317016515414327, −3.02515313148462909596280079041, −2.40088803603320759065892897292, −1.14474566370925053529227145366, 1.14474566370925053529227145366, 2.40088803603320759065892897292, 3.02515313148462909596280079041, 4.04578995504124317016515414327, 4.83969109724384096480112467617, 6.07102379371770908331786128296, 6.74168988426297517090799028191, 7.44408125712967954118004582988, 8.608338612079200641754060102542, 8.827440748797551310744489617320

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