Properties

Label 2-3380-1.1-c1-0-7
Degree $2$
Conductor $3380$
Sign $1$
Analytic cond. $26.9894$
Root an. cond. $5.19513$
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  − 3.40·3-s − 5-s − 2.33·7-s + 8.60·9-s + 6.16·11-s + 3.40·15-s + 3.07·17-s + 0.0902·19-s + 7.96·21-s + 4.81·23-s + 25-s − 19.0·27-s + 5.63·29-s − 8.34·31-s − 21.0·33-s + 2.33·35-s − 0.794·37-s + 3.95·41-s − 8.75·43-s − 8.60·45-s + 3.67·47-s − 1.53·49-s − 10.4·51-s + 8.08·53-s − 6.16·55-s − 0.307·57-s − 0.379·59-s + ⋯
L(s)  = 1  − 1.96·3-s − 0.447·5-s − 0.883·7-s + 2.86·9-s + 1.85·11-s + 0.879·15-s + 0.745·17-s + 0.0207·19-s + 1.73·21-s + 1.00·23-s + 0.200·25-s − 3.67·27-s + 1.04·29-s − 1.49·31-s − 3.65·33-s + 0.395·35-s − 0.130·37-s + 0.618·41-s − 1.33·43-s − 1.28·45-s + 0.536·47-s − 0.219·49-s − 1.46·51-s + 1.11·53-s − 0.831·55-s − 0.0407·57-s − 0.0494·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3380\)    =    \(2^{2} \cdot 5 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(26.9894\)
Root analytic conductor: \(5.19513\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3380} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3380,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8196668144\)
\(L(\frac12)\) \(\approx\) \(0.8196668144\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
13 \( 1 \)
good3 \( 1 + 3.40T + 3T^{2} \)
7 \( 1 + 2.33T + 7T^{2} \)
11 \( 1 - 6.16T + 11T^{2} \)
17 \( 1 - 3.07T + 17T^{2} \)
19 \( 1 - 0.0902T + 19T^{2} \)
23 \( 1 - 4.81T + 23T^{2} \)
29 \( 1 - 5.63T + 29T^{2} \)
31 \( 1 + 8.34T + 31T^{2} \)
37 \( 1 + 0.794T + 37T^{2} \)
41 \( 1 - 3.95T + 41T^{2} \)
43 \( 1 + 8.75T + 43T^{2} \)
47 \( 1 - 3.67T + 47T^{2} \)
53 \( 1 - 8.08T + 53T^{2} \)
59 \( 1 + 0.379T + 59T^{2} \)
61 \( 1 + 4.96T + 61T^{2} \)
67 \( 1 + 0.376T + 67T^{2} \)
71 \( 1 - 9.04T + 71T^{2} \)
73 \( 1 + 5.48T + 73T^{2} \)
79 \( 1 + 4.79T + 79T^{2} \)
83 \( 1 + 11.6T + 83T^{2} \)
89 \( 1 - 11.7T + 89T^{2} \)
97 \( 1 + 2.43T + 97T^{2} \)
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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

−8.797975555858564994521396781086, −7.42742584525047477961042060176, −6.89886167555757737212228960119, −6.37869982901242399345846756468, −5.71885275653110581456076763228, −4.88341278241790198948625739816, −4.06970216895660769464798529052, −3.38911233934669049877242905290, −1.49609864538553886379069701153, −0.65170403552692261229580577739, 0.65170403552692261229580577739, 1.49609864538553886379069701153, 3.38911233934669049877242905290, 4.06970216895660769464798529052, 4.88341278241790198948625739816, 5.71885275653110581456076763228, 6.37869982901242399345846756468, 6.89886167555757737212228960119, 7.42742584525047477961042060176, 8.797975555858564994521396781086

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