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Properties

Label 2-1560-1560.1403-c0-0-6
Degree $2$
Conductor $1560$
Sign $0.229 + 0.973i$
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 + (−0.5 + 0.866i)3-s + 1.00i·4-s + (−0.965 − 0.258i)5-s + (0.965 − 0.258i)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 + (−0.866 − 0.500i)12-s + (−0.707 − 0.707i)13-s + 1.73·14-s + (0.707 − 0.707i)15-s − 1.00·16-s + (−0.366 − 0.366i)17-s + (−0.258 + 0.965i)18-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + (−0.5 + 0.866i)3-s + 1.00i·4-s + (−0.965 − 0.258i)5-s + (0.965 − 0.258i)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 + (−0.866 − 0.500i)12-s + (−0.707 − 0.707i)13-s + 1.73·14-s + (0.707 − 0.707i)15-s − 1.00·16-s + (−0.366 − 0.366i)17-s + (−0.258 + 0.965i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1560\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.229 + 0.973i$
Analytic conductor: \(0.778541\)
Root analytic conductor: \(0.882349\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1560} (1403, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1560,\ (\ :0),\ 0.229 + 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2508892361\)
\(L(\frac12)\) \(\approx\) \(0.2508892361\)
\(L(1)\) 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 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (0.965 + 0.258i)T \)
13 \( 1 + (0.707 + 0.707i)T \)
good7 \( 1 + (1.22 - 1.22i)T - iT^{2} \)
11 \( 1 + T^{2} \)
17 \( 1 + (0.366 + 0.366i)T + iT^{2} \)
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 \( 1 + (1.36 + 1.36i)T + iT^{2} \)
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 \( 1 + 1.93iT - T^{2} \)
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

−9.492939663466584911317802781285, −8.924381518418183991234583780646, −8.211978454130485024358988711243, −7.16620615272404515806460097240, −6.22176428299551150821333765458, −5.16386838129367475050927464789, −4.25034984568747819746263077057, −3.25239770704115588825129588254, −2.65007174436710436220478710794, −0.33771820937268692910193092819, 0.991281592294065864685394546355, 2.64647866243250958844298617214, 4.05934879015453388052614133463, 4.93472065335988754865192816582, 6.38343384193938593128807318054, 6.62232055153585748126703186344, 7.33610321080108248008391726431, 7.924810111034153616537701601952, 8.784627539648574389151069453813, 9.945188298261589673775880616515

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