Properties

Label 2-4020-5.4-c1-0-19
Degree $2$
Conductor $4020$
Sign $-0.650 - 0.759i$
Analytic cond. $32.0998$
Root an. cond. $5.66567$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·3-s + (1.45 + 1.69i)5-s + 0.243i·7-s − 9-s + 0.348·11-s − 2.62i·13-s + (−1.69 + 1.45i)15-s − 1.29i·17-s + 1.36·19-s − 0.243·21-s + 0.600i·23-s + (−0.768 + 4.94i)25-s i·27-s − 3.99·29-s − 3.23·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + (0.650 + 0.759i)5-s + 0.0919i·7-s − 0.333·9-s + 0.105·11-s − 0.728i·13-s + (−0.438 + 0.375i)15-s − 0.313i·17-s + 0.313·19-s − 0.0530·21-s + 0.125i·23-s + (−0.153 + 0.988i)25-s − 0.192i·27-s − 0.742·29-s − 0.581·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.650 - 0.759i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.650 - 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
Sign: $-0.650 - 0.759i$
Analytic conductor: \(32.0998\)
Root analytic conductor: \(5.66567\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4020} (1609, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4020,\ (\ :1/2),\ -0.650 - 0.759i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.674007456\)
\(L(\frac12)\) \(\approx\) \(1.674007456\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
5 \( 1 + (-1.45 - 1.69i)T \)
67 \( 1 + iT \)
good7 \( 1 - 0.243iT - 7T^{2} \)
11 \( 1 - 0.348T + 11T^{2} \)
13 \( 1 + 2.62iT - 13T^{2} \)
17 \( 1 + 1.29iT - 17T^{2} \)
19 \( 1 - 1.36T + 19T^{2} \)
23 \( 1 - 0.600iT - 23T^{2} \)
29 \( 1 + 3.99T + 29T^{2} \)
31 \( 1 + 3.23T + 31T^{2} \)
37 \( 1 - 7.96iT - 37T^{2} \)
41 \( 1 - 2.27T + 41T^{2} \)
43 \( 1 - 8.24iT - 43T^{2} \)
47 \( 1 - 9.05iT - 47T^{2} \)
53 \( 1 - 10.5iT - 53T^{2} \)
59 \( 1 - 6.00T + 59T^{2} \)
61 \( 1 + 3.78T + 61T^{2} \)
71 \( 1 + 4.45T + 71T^{2} \)
73 \( 1 - 8.87iT - 73T^{2} \)
79 \( 1 + 16.2T + 79T^{2} \)
83 \( 1 + 10.3iT - 83T^{2} \)
89 \( 1 - 8.64T + 89T^{2} \)
97 \( 1 + 0.828iT - 97T^{2} \)
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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

−8.946360779214847982440811266421, −7.86164980833785124065103902545, −7.32858802830909041655525329031, −6.34395510939961900539388920863, −5.77177593930676452609797995188, −5.06801731818941028265367042498, −4.11013977887788071844304438004, −3.14148779880723777450476326238, −2.60466474445421424270572958765, −1.31581286001372227677216047770, 0.47455804217907918766702985701, 1.68970625266392880859543982205, 2.25280534915866813532777022035, 3.60246876792657419066380403355, 4.38545633426045524540569733930, 5.44491991927545008833871406525, 5.79772923783197995289712339993, 6.83588300811870067518277354373, 7.30916602486203131698250849964, 8.305795872361121820527203268687

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