Properties

Label 2-1560-8.5-c1-0-87
Degree $2$
Conductor $1560$
Sign $i$
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  + 1.41·2-s i·3-s + 2.00·4-s i·5-s − 1.41i·6-s − 0.585·7-s + 2.82·8-s − 9-s − 1.41i·10-s − 4.82i·11-s − 2.00i·12-s + i·13-s − 0.828·14-s − 15-s + 4.00·16-s − 1.41·17-s + ⋯
L(s)  = 1  + 1.00·2-s − 0.577i·3-s + 1.00·4-s − 0.447i·5-s − 0.577i·6-s − 0.221·7-s + 1.00·8-s − 0.333·9-s − 0.447i·10-s − 1.45i·11-s − 0.577i·12-s + 0.277i·13-s − 0.221·14-s − 0.258·15-s + 1.00·16-s − 0.342·17-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1560\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 13\)
Sign: $i$
Analytic conductor: \(12.4566\)
Root analytic conductor: \(3.52939\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1560} (781, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1560,\ (\ :1/2),\ i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.114565326\)
\(L(\frac12)\) \(\approx\) \(3.114565326\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41T \)
3 \( 1 + iT \)
5 \( 1 + iT \)
13 \( 1 - iT \)
good7 \( 1 + 0.585T + 7T^{2} \)
11 \( 1 + 4.82iT - 11T^{2} \)
17 \( 1 + 1.41T + 17T^{2} \)
19 \( 1 - 0.242iT - 19T^{2} \)
23 \( 1 - 4.24T + 23T^{2} \)
29 \( 1 + 10.2iT - 29T^{2} \)
31 \( 1 - 1.17T + 31T^{2} \)
37 \( 1 + 6.48iT - 37T^{2} \)
41 \( 1 + 5.17T + 41T^{2} \)
43 \( 1 - 4.82iT - 43T^{2} \)
47 \( 1 - 1.65T + 47T^{2} \)
53 \( 1 - 0.343iT - 53T^{2} \)
59 \( 1 - 3.17iT - 59T^{2} \)
61 \( 1 - 8iT - 61T^{2} \)
67 \( 1 - 1.17iT - 67T^{2} \)
71 \( 1 - 4.82T + 71T^{2} \)
73 \( 1 - 8.58T + 73T^{2} \)
79 \( 1 + 1.65T + 79T^{2} \)
83 \( 1 - 7.65iT - 83T^{2} \)
89 \( 1 - 7.65T + 89T^{2} \)
97 \( 1 - 6.72T + 97T^{2} \)
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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

−9.100089526109472335180481161653, −8.279745097271503458386430128080, −7.54313293225158278404930462605, −6.53111915405996632815655851254, −6.00332504028517107001877758701, −5.17790722183384553129135132253, −4.14967293699980985124666823655, −3.21610395328258209668244637367, −2.23592910235349755036143940126, −0.871538998233061268934599877208, 1.79837612912412369149625355229, 2.93268023043620584814641290223, 3.67072693385860023285196093991, 4.80152857425384719392127801309, 5.15391830942896605490556932038, 6.47455221693267939520418297351, 6.93572540300142579304872767186, 7.80858676638947459315516554346, 8.941375719210057985775715721832, 9.897771754022871608683271297414

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