Properties

Label 2-4080-5.4-c1-0-3
Degree $2$
Conductor $4080$
Sign $-i$
Analytic cond. $32.5789$
Root an. cond. $5.70779$
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 − 2.23i·5-s + 1.23i·7-s − 9-s − 2·11-s − 5.23i·13-s − 2.23·15-s + i·17-s − 6.47·19-s + 1.23·21-s + 4i·23-s − 5.00·25-s + i·27-s + 4·29-s − 2.47·31-s + ⋯
L(s)  = 1  − 0.577i·3-s − 0.999i·5-s + 0.467i·7-s − 0.333·9-s − 0.603·11-s − 1.45i·13-s − 0.577·15-s + 0.242i·17-s − 1.48·19-s + 0.269·21-s + 0.834i·23-s − 1.00·25-s + 0.192i·27-s + 0.742·29-s − 0.444·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4080\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 17\)
Sign: $-i$
Analytic conductor: \(32.5789\)
Root analytic conductor: \(5.70779\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4080} (2449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4080,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2819479959\)
\(L(\frac12)\) \(\approx\) \(0.2819479959\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.23iT \)
17 \( 1 - iT \)
good7 \( 1 - 1.23iT - 7T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 + 5.23iT - 13T^{2} \)
19 \( 1 + 6.47T + 19T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 + 2.47T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + 7.70T + 41T^{2} \)
43 \( 1 - 12.1iT - 43T^{2} \)
47 \( 1 + 10.4iT - 47T^{2} \)
53 \( 1 - 10.9iT - 53T^{2} \)
59 \( 1 - 11.7T + 59T^{2} \)
61 \( 1 - 4.47T + 61T^{2} \)
67 \( 1 - 1.70iT - 67T^{2} \)
71 \( 1 + 11.2T + 71T^{2} \)
73 \( 1 + 8.76iT - 73T^{2} \)
79 \( 1 - 0.944T + 79T^{2} \)
83 \( 1 - 10.4iT - 83T^{2} \)
89 \( 1 + 12.4T + 89T^{2} \)
97 \( 1 + 1.70iT - 97T^{2} \)
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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

−8.356268082747924537089856823767, −8.154217623579506587427544614520, −7.27401757610306078317800045097, −6.28968987205691368975934624584, −5.59072527763094925640632060388, −5.09300375656129953604384451321, −4.10610048722845622415751108265, −3.03120219789291241890020678953, −2.15056542184496250130093119113, −1.10301304985688485936536160564, 0.082846234740168725565072584165, 1.98247643508177530252271915590, 2.67782010055908602708073282445, 3.79966083414469113715176292498, 4.27089181925853376177795912031, 5.17843912155961614641938952091, 6.19242628248492608153194271865, 6.81071349077508861342855292016, 7.32066156544698011912306538269, 8.434058776989840827254572558587

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