Properties

Label 2-4020-1.1-c1-0-23
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s + 1.68·7-s + 9-s + 6.11·11-s + 4.10·13-s − 15-s + 1.36·17-s + 4.20·19-s + 1.68·21-s − 5.55·23-s + 25-s + 27-s + 9.42·29-s − 0.853·31-s + 6.11·33-s − 1.68·35-s − 7.11·37-s + 4.10·39-s − 3.25·41-s − 9.08·43-s − 45-s − 3.42·47-s − 4.16·49-s + 1.36·51-s + 10.9·53-s − 6.11·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 0.636·7-s + 0.333·9-s + 1.84·11-s + 1.13·13-s − 0.258·15-s + 0.330·17-s + 0.965·19-s + 0.367·21-s − 1.15·23-s + 0.200·25-s + 0.192·27-s + 1.75·29-s − 0.153·31-s + 1.06·33-s − 0.284·35-s − 1.16·37-s + 0.658·39-s − 0.507·41-s − 1.38·43-s − 0.149·45-s − 0.499·47-s − 0.594·49-s + 0.190·51-s + 1.50·53-s − 0.824·55-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: \(0\)
Selberg data: \((2,\ 4020,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.021859127\)
\(L(\frac12)\) \(\approx\) \(3.021859127\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.68T + 7T^{2} \)
11 \( 1 - 6.11T + 11T^{2} \)
13 \( 1 - 4.10T + 13T^{2} \)
17 \( 1 - 1.36T + 17T^{2} \)
19 \( 1 - 4.20T + 19T^{2} \)
23 \( 1 + 5.55T + 23T^{2} \)
29 \( 1 - 9.42T + 29T^{2} \)
31 \( 1 + 0.853T + 31T^{2} \)
37 \( 1 + 7.11T + 37T^{2} \)
41 \( 1 + 3.25T + 41T^{2} \)
43 \( 1 + 9.08T + 43T^{2} \)
47 \( 1 + 3.42T + 47T^{2} \)
53 \( 1 - 10.9T + 53T^{2} \)
59 \( 1 - 4.21T + 59T^{2} \)
61 \( 1 + 4.10T + 61T^{2} \)
71 \( 1 - 7.68T + 71T^{2} \)
73 \( 1 + 15.3T + 73T^{2} \)
79 \( 1 - 16.1T + 79T^{2} \)
83 \( 1 + 0.430T + 83T^{2} \)
89 \( 1 + 18.3T + 89T^{2} \)
97 \( 1 - 9.36T + 97T^{2} \)
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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

−8.443364007941354519899264323352, −7.937330449628553677796339917724, −6.90645742409711824566110298784, −6.46559450016778587494743122376, −5.43159662351146387353664030397, −4.46453660803788916742163592015, −3.76402891205986789806539529261, −3.20982438232643432281092794971, −1.76687251632634790541224133387, −1.09834383463526026182332243961, 1.09834383463526026182332243961, 1.76687251632634790541224133387, 3.20982438232643432281092794971, 3.76402891205986789806539529261, 4.46453660803788916742163592015, 5.43159662351146387353664030397, 6.46559450016778587494743122376, 6.90645742409711824566110298784, 7.937330449628553677796339917724, 8.443364007941354519899264323352

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