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Properties

Label 2-1560-1560.467-c0-0-14
Degree $2$
Conductor $1560$
Sign $-0.685 + 0.727i$
Analytic cond. $0.778541$
Root an. cond. $0.882349$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 0.707i)2-s + (−0.866 + 0.5i)3-s − 1.00i·4-s + (0.258 − 0.965i)5-s + (0.258 − 0.965i)6-s + (−1.22 − 1.22i)7-s + (0.707 + 0.707i)8-s + (0.499 − 0.866i)9-s + (0.500 + 0.866i)10-s + (0.500 + 0.866i)12-s + (0.707 − 0.707i)13-s + 1.73·14-s + (0.258 + 0.965i)15-s − 1.00·16-s + (−1.36 + 1.36i)17-s + (0.258 + 0.965i)18-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (−0.866 + 0.5i)3-s − 1.00i·4-s + (0.258 − 0.965i)5-s + (0.258 − 0.965i)6-s + (−1.22 − 1.22i)7-s + (0.707 + 0.707i)8-s + (0.499 − 0.866i)9-s + (0.500 + 0.866i)10-s + (0.500 + 0.866i)12-s + (0.707 − 0.707i)13-s + 1.73·14-s + (0.258 + 0.965i)15-s − 1.00·16-s + (−1.36 + 1.36i)17-s + (0.258 + 0.965i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.685 + 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.685 + 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1560\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.685 + 0.727i$
Analytic conductor: \(0.778541\)
Root analytic conductor: \(0.882349\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1560} (467, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1560,\ (\ :0),\ -0.685 + 0.727i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1889417668\)
\(L(\frac12)\) \(\approx\) \(0.1889417668\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 + (-0.258 + 0.965i)T \)
13 \( 1 + (-0.707 + 0.707i)T \)
good7 \( 1 + (1.22 + 1.22i)T + iT^{2} \)
11 \( 1 + T^{2} \)
17 \( 1 + (1.36 - 1.36i)T - iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + 1.41T + T^{2} \)
37 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (-0.366 + 0.366i)T - iT^{2} \)
47 \( 1 + (1.22 - 1.22i)T - iT^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + 0.517iT - T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - iT^{2} \)
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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

−9.314156394908595983217664962678, −8.740590277374448862095208124323, −7.70381853521301399989945357776, −6.72731613029959101705225118458, −6.17935487714673840901843422222, −5.47513238400709685488707739802, −4.40625191988705996659871501937, −3.71414211244370059451699083740, −1.49445078338967690324858197497, −0.20938401333422150411691943233, 1.92860779294310448984809269966, 2.66401420579489821608268820434, 3.67525505282066173698859503501, 5.07609862328701178739803568744, 6.21378039416327216439320269639, 6.72021301413702601418885603235, 7.30656540929053864664443144615, 8.609875779345686692446247726665, 9.277940023054684223910115172105, 9.915098115038396683773666851808

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