Properties

Label 2-1560-5.4-c1-0-5
Degree $2$
Conductor $1560$
Sign $-0.975 + 0.218i$
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 + (0.489 + 2.18i)5-s + 2.82i·7-s − 9-s − 2.97·11-s i·13-s + (−2.18 + 0.489i)15-s − 1.44i·17-s − 4.17·19-s − 2.82·21-s + 6.21i·23-s + (−4.52 + 2.13i)25-s i·27-s + 0.828·29-s + 1.65·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + (0.218 + 0.975i)5-s + 1.06i·7-s − 0.333·9-s − 0.898·11-s − 0.277i·13-s + (−0.563 + 0.126i)15-s − 0.350i·17-s − 0.956·19-s − 0.617·21-s + 1.29i·23-s + (−0.904 + 0.427i)25-s − 0.192i·27-s + 0.153·29-s + 0.297·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 + 0.218i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.975 + 0.218i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1560\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.975 + 0.218i$
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: \((2,\ 1560,\ (\ :1/2),\ -0.975 + 0.218i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8823770038\)
\(L(\frac12)\) \(\approx\) \(0.8823770038\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.489 - 2.18i)T \)
13 \( 1 + iT \)
good7 \( 1 - 2.82iT - 7T^{2} \)
11 \( 1 + 2.97T + 11T^{2} \)
17 \( 1 + 1.44iT - 17T^{2} \)
19 \( 1 + 4.17T + 19T^{2} \)
23 \( 1 - 6.21iT - 23T^{2} \)
29 \( 1 - 0.828T + 29T^{2} \)
31 \( 1 - 1.65T + 31T^{2} \)
37 \( 1 - 0.0418iT - 37T^{2} \)
41 \( 1 + 4.36T + 41T^{2} \)
43 \( 1 + 8.99iT - 43T^{2} \)
47 \( 1 - 4.40iT - 47T^{2} \)
53 \( 1 + 6.72iT - 53T^{2} \)
59 \( 1 - 2.97T + 59T^{2} \)
61 \( 1 - 6.27T + 61T^{2} \)
67 \( 1 + 0.727iT - 67T^{2} \)
71 \( 1 + 8.32T + 71T^{2} \)
73 \( 1 + 6.87iT - 73T^{2} \)
79 \( 1 + 6.11T + 79T^{2} \)
83 \( 1 - 14.7iT - 83T^{2} \)
89 \( 1 + 14.7T + 89T^{2} \)
97 \( 1 - 6.78iT - 97T^{2} \)
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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

−9.983544238742181955562336404790, −9.127770697041665496707415087174, −8.354393127973993433661746857960, −7.49965213362082306999576865106, −6.54096419665004333296573981415, −5.65851981098042855512567642756, −5.12466758967972978348186009545, −3.80394162690211273253729243355, −2.87126455673433378858816662331, −2.13898418732956772882879620457, 0.32795254603520698790392172039, 1.52570592846523494105127046913, 2.67335793089738469421783460881, 4.12110567199663120261721839597, 4.72620960278122994278930763150, 5.79161889354010319351125951481, 6.61871099634305029664909615412, 7.45537558029780828748033550904, 8.309106573704410257395299900194, 8.721412693735118153290887169638

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