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Properties

Label 2-560-35.12-c1-0-19
Degree $2$
Conductor $560$
Sign $-0.556 + 0.830i$
Analytic cond. $4.47162$
Root an. cond. $2.11462$
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  + (0.364 − 1.35i)3-s + (−0.489 − 2.18i)5-s + (0.501 − 2.59i)7-s + (0.882 + 0.509i)9-s + (−1.86 − 3.23i)11-s + (4.55 + 4.55i)13-s + (−3.14 − 0.129i)15-s + (−5.91 − 1.58i)17-s + (−0.616 + 1.06i)19-s + (−3.34 − 1.62i)21-s + (−0.721 − 2.69i)23-s + (−4.52 + 2.13i)25-s + (3.99 − 3.99i)27-s − 2.82i·29-s + (−5.94 + 3.43i)31-s + ⋯
L(s)  = 1  + (0.210 − 0.784i)3-s + (−0.218 − 0.975i)5-s + (0.189 − 0.981i)7-s + (0.294 + 0.169i)9-s + (−0.563 − 0.975i)11-s + (1.26 + 1.26i)13-s + (−0.811 − 0.0333i)15-s + (−1.43 − 0.384i)17-s + (−0.141 + 0.244i)19-s + (−0.730 − 0.355i)21-s + (−0.150 − 0.561i)23-s + (−0.904 + 0.427i)25-s + (0.769 − 0.769i)27-s − 0.525i·29-s + (−1.06 + 0.616i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.556 + 0.830i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.556 + 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $-0.556 + 0.830i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ -0.556 + 0.830i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.673407 - 1.26170i\)
\(L(\frac12)\) \(\approx\) \(0.673407 - 1.26170i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.489 + 2.18i)T \)
7 \( 1 + (-0.501 + 2.59i)T \)
good3 \( 1 + (-0.364 + 1.35i)T + (-2.59 - 1.5i)T^{2} \)
11 \( 1 + (1.86 + 3.23i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.55 - 4.55i)T + 13iT^{2} \)
17 \( 1 + (5.91 + 1.58i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (0.616 - 1.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.721 + 2.69i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 2.82iT - 29T^{2} \)
31 \( 1 + (5.94 - 3.43i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-7.52 + 2.01i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 5.56iT - 41T^{2} \)
43 \( 1 + (-1.95 + 1.95i)T - 43iT^{2} \)
47 \( 1 + (-2.98 - 11.1i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-9.44 - 2.53i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (0.916 + 1.58i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.43 + 1.98i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.12 + 7.93i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 0.570T + 71T^{2} \)
73 \( 1 + (0.546 - 2.03i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-9.47 - 5.47i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.80 - 8.80i)T + 83iT^{2} \)
89 \( 1 + (-3.00 + 5.20i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.70 - 3.70i)T - 97iT^{2} \)
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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

−10.78935814032727147060270077501, −9.314923375808492284513766506690, −8.584180159834799575619368960435, −7.85009200552328164216456847082, −6.94220832954956373169309967701, −6.02411939818581042007244546732, −4.59515186688801271612834659524, −3.90978434274318065342030742957, −2.04492374462247140617967779348, −0.827668920799750366212844267095, 2.21524219365472038619152662960, 3.33921763387174637675775921032, 4.33066962311554817757035930389, 5.51574261989750012739326821530, 6.48770445820754434733798856083, 7.55511719510310076147228012906, 8.522277896427777394404259200461, 9.372240356397870623211821893456, 10.31146088404891920533748596416, 10.86554789666452024390966165850

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