Properties

Label 2-6020-1.1-c1-0-76
Degree $2$
Conductor $6020$
Sign $-1$
Analytic cond. $48.0699$
Root an. cond. $6.93324$
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  + 1.96·3-s − 5-s + 7-s + 0.865·9-s + 2.19·11-s − 5.77·13-s − 1.96·15-s − 1.81·17-s − 3.83·19-s + 1.96·21-s + 5.76·23-s + 25-s − 4.19·27-s − 0.573·29-s + 4.79·31-s + 4.30·33-s − 35-s − 4.95·37-s − 11.3·39-s − 7.21·41-s − 43-s − 0.865·45-s − 11.0·47-s + 49-s − 3.56·51-s + 5.76·53-s − 2.19·55-s + ⋯
L(s)  = 1  + 1.13·3-s − 0.447·5-s + 0.377·7-s + 0.288·9-s + 0.660·11-s − 1.60·13-s − 0.507·15-s − 0.439·17-s − 0.879·19-s + 0.429·21-s + 1.20·23-s + 0.200·25-s − 0.807·27-s − 0.106·29-s + 0.861·31-s + 0.750·33-s − 0.169·35-s − 0.814·37-s − 1.81·39-s − 1.12·41-s − 0.152·43-s − 0.129·45-s − 1.61·47-s + 0.142·49-s − 0.499·51-s + 0.791·53-s − 0.295·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6020\)    =    \(2^{2} \cdot 5 \cdot 7 \cdot 43\)
Sign: $-1$
Analytic conductor: \(48.0699\)
Root analytic conductor: \(6.93324\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6020,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
7 \( 1 - T \)
43 \( 1 + T \)
good3 \( 1 - 1.96T + 3T^{2} \)
11 \( 1 - 2.19T + 11T^{2} \)
13 \( 1 + 5.77T + 13T^{2} \)
17 \( 1 + 1.81T + 17T^{2} \)
19 \( 1 + 3.83T + 19T^{2} \)
23 \( 1 - 5.76T + 23T^{2} \)
29 \( 1 + 0.573T + 29T^{2} \)
31 \( 1 - 4.79T + 31T^{2} \)
37 \( 1 + 4.95T + 37T^{2} \)
41 \( 1 + 7.21T + 41T^{2} \)
47 \( 1 + 11.0T + 47T^{2} \)
53 \( 1 - 5.76T + 53T^{2} \)
59 \( 1 + 0.651T + 59T^{2} \)
61 \( 1 - 0.841T + 61T^{2} \)
67 \( 1 - 4.14T + 67T^{2} \)
71 \( 1 + 4.87T + 71T^{2} \)
73 \( 1 - 11.0T + 73T^{2} \)
79 \( 1 + 8.10T + 79T^{2} \)
83 \( 1 + 12.0T + 83T^{2} \)
89 \( 1 + 16.6T + 89T^{2} \)
97 \( 1 + 8.76T + 97T^{2} \)
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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

−7.87136187744633353173731927228, −7.03622656214912459234642814028, −6.67534706994541294112992084184, −5.38192005650868975558472715123, −4.69222023011357353153300020628, −4.00040369869186371856399275906, −3.10222773977841191498270757047, −2.46500979569571257468403309597, −1.57039528009365363449734863856, 0, 1.57039528009365363449734863856, 2.46500979569571257468403309597, 3.10222773977841191498270757047, 4.00040369869186371856399275906, 4.69222023011357353153300020628, 5.38192005650868975558472715123, 6.67534706994541294112992084184, 7.03622656214912459234642814028, 7.87136187744633353173731927228

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