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Properties

Label 2-1560-5.4-c1-0-14
Degree $2$
Conductor $1560$
Sign $0.844 - 0.535i$
Analytic cond. $12.4566$
Root an. cond. $3.52939$
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 + (−1.19 − 1.88i)5-s + 2.82i·7-s − 9-s + 0.393·11-s − i·13-s + (1.88 − 1.19i)15-s − 6.21i·17-s + 7.34·19-s − 2.82·21-s + 1.44i·23-s + (−2.13 + 4.52i)25-s − i·27-s + 0.828·29-s + 1.65·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + (−0.535 − 0.844i)5-s + 1.06i·7-s − 0.333·9-s + 0.118·11-s − 0.277i·13-s + (0.487 − 0.308i)15-s − 1.50i·17-s + 1.68·19-s − 0.617·21-s + 0.301i·23-s + (−0.427 + 0.904i)25-s − 0.192i·27-s + 0.153·29-s + 0.297·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 - 0.535i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.844 - 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1560\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.844 - 0.535i$
Analytic conductor: \(12.4566\)
Root analytic conductor: \(3.52939\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1560} (1249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1560,\ (\ :1/2),\ 0.844 - 0.535i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.514524268\)
\(L(\frac12)\) \(\approx\) \(1.514524268\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.19 + 1.88i)T \)
13 \( 1 + iT \)
good7 \( 1 - 2.82iT - 7T^{2} \)
11 \( 1 - 0.393T + 11T^{2} \)
17 \( 1 + 6.21iT - 17T^{2} \)
19 \( 1 - 7.34T + 19T^{2} \)
23 \( 1 - 1.44iT - 23T^{2} \)
29 \( 1 - 0.828T + 29T^{2} \)
31 \( 1 - 1.65T + 31T^{2} \)
37 \( 1 - 6.78iT - 37T^{2} \)
41 \( 1 - 3.77T + 41T^{2} \)
43 \( 1 - 2.51iT - 43T^{2} \)
47 \( 1 - 3.00iT - 47T^{2} \)
53 \( 1 - 9.55iT - 53T^{2} \)
59 \( 1 + 0.393T + 59T^{2} \)
61 \( 1 - 11.0T + 61T^{2} \)
67 \( 1 - 15.5iT - 67T^{2} \)
71 \( 1 - 6.56T + 71T^{2} \)
73 \( 1 + 13.6iT - 73T^{2} \)
79 \( 1 - 14.9T + 79T^{2} \)
83 \( 1 + 16.4iT - 83T^{2} \)
89 \( 1 - 9.67T + 89T^{2} \)
97 \( 1 - 0.0418iT - 97T^{2} \)
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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

−9.335085274910427305131338583986, −8.943351934776913671598995192116, −7.989533383972435431094855761447, −7.31173322925030056336138720417, −5.99760270435034360899328272121, −5.19067454313575150573731981519, −4.72980084130521870132613349675, −3.47966914888879384451106497508, −2.64434246040363223438481490201, −0.959980320387941869754719220823, 0.819295712751192257731995677036, 2.18907882736173836328995768789, 3.49555492351764625154428319522, 3.99615017152054628247184647159, 5.32340419944529965841065712418, 6.42127077039447582069007626487, 6.96726801603892054113298035055, 7.71245999452670598898556053826, 8.255385340887474081418610165750, 9.441121417004119848500425157470

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