Properties

Label 2-4020-1.1-c1-0-42
Degree $2$
Conductor $4020$
Sign $-1$
Analytic cond. $32.0998$
Root an. cond. $5.66567$
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  + 3-s + 5-s − 1.40·7-s + 9-s + 3.19·11-s − 7.12·13-s + 15-s + 5.07·17-s − 5.01·19-s − 1.40·21-s − 3.97·23-s + 25-s + 27-s − 9.15·29-s − 7.41·31-s + 3.19·33-s − 1.40·35-s − 5.97·37-s − 7.12·39-s + 0.574·41-s − 0.608·43-s + 45-s + 2.70·47-s − 5.01·49-s + 5.07·51-s + 3.64·53-s + 3.19·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 0.532·7-s + 0.333·9-s + 0.962·11-s − 1.97·13-s + 0.258·15-s + 1.23·17-s − 1.15·19-s − 0.307·21-s − 0.829·23-s + 0.200·25-s + 0.192·27-s − 1.70·29-s − 1.33·31-s + 0.555·33-s − 0.238·35-s − 0.982·37-s − 1.14·39-s + 0.0896·41-s − 0.0928·43-s + 0.149·45-s + 0.394·47-s − 0.716·49-s + 0.711·51-s + 0.500·53-s + 0.430·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
Sign: $-1$
Analytic conductor: \(32.0998\)
Root analytic conductor: \(5.66567\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4020,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 - T \)
5 \( 1 - T \)
67 \( 1 - T \)
good7 \( 1 + 1.40T + 7T^{2} \)
11 \( 1 - 3.19T + 11T^{2} \)
13 \( 1 + 7.12T + 13T^{2} \)
17 \( 1 - 5.07T + 17T^{2} \)
19 \( 1 + 5.01T + 19T^{2} \)
23 \( 1 + 3.97T + 23T^{2} \)
29 \( 1 + 9.15T + 29T^{2} \)
31 \( 1 + 7.41T + 31T^{2} \)
37 \( 1 + 5.97T + 37T^{2} \)
41 \( 1 - 0.574T + 41T^{2} \)
43 \( 1 + 0.608T + 43T^{2} \)
47 \( 1 - 2.70T + 47T^{2} \)
53 \( 1 - 3.64T + 53T^{2} \)
59 \( 1 - 4.84T + 59T^{2} \)
61 \( 1 + 0.942T + 61T^{2} \)
71 \( 1 - 4.36T + 71T^{2} \)
73 \( 1 + 9.56T + 73T^{2} \)
79 \( 1 + 6.26T + 79T^{2} \)
83 \( 1 - 2.76T + 83T^{2} \)
89 \( 1 - 17.4T + 89T^{2} \)
97 \( 1 - 7.08T + 97T^{2} \)
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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

−8.008004080651893172003956518427, −7.33397127875288911481067103542, −6.76478545684822891746811015603, −5.83158315022303789648151257065, −5.14188644371584092288033968879, −4.08256926417101592944951646741, −3.44553612205098726120072835516, −2.38016350821095152849849696507, −1.69889195859689866806536480943, 0, 1.69889195859689866806536480943, 2.38016350821095152849849696507, 3.44553612205098726120072835516, 4.08256926417101592944951646741, 5.14188644371584092288033968879, 5.83158315022303789648151257065, 6.76478545684822891746811015603, 7.33397127875288911481067103542, 8.008004080651893172003956518427

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