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Properties

Label 2-560-112.109-c1-0-18
Degree $2$
Conductor $560$
Sign $-0.663 - 0.748i$
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  + (0.313 + 1.37i)2-s + (0.215 − 0.0577i)3-s + (−1.80 + 0.864i)4-s + (0.965 + 0.258i)5-s + (0.147 + 0.279i)6-s + (0.618 − 2.57i)7-s + (−1.75 − 2.21i)8-s + (−2.55 + 1.47i)9-s + (−0.0541 + 1.41i)10-s + (1.49 + 5.56i)11-s + (−0.338 + 0.290i)12-s + (2.05 + 2.05i)13-s + (3.74 + 0.0466i)14-s + 0.223·15-s + (2.50 − 3.11i)16-s + (−2.61 + 4.53i)17-s + ⋯
L(s)  = 1  + (0.221 + 0.975i)2-s + (0.124 − 0.0333i)3-s + (−0.901 + 0.432i)4-s + (0.431 + 0.115i)5-s + (0.0601 + 0.114i)6-s + (0.233 − 0.972i)7-s + (−0.621 − 0.783i)8-s + (−0.851 + 0.491i)9-s + (−0.0171 + 0.446i)10-s + (0.449 + 1.67i)11-s + (−0.0978 + 0.0838i)12-s + (0.569 + 0.569i)13-s + (0.999 + 0.0124i)14-s + 0.0576·15-s + (0.626 − 0.779i)16-s + (−0.635 + 1.10i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.663 - 0.748i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.663 - 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $-0.663 - 0.748i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ -0.663 - 0.748i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.605684 + 1.34666i\)
\(L(\frac12)\) \(\approx\) \(0.605684 + 1.34666i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.313 - 1.37i)T \)
5 \( 1 + (-0.965 - 0.258i)T \)
7 \( 1 + (-0.618 + 2.57i)T \)
good3 \( 1 + (-0.215 + 0.0577i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (-1.49 - 5.56i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (-2.05 - 2.05i)T + 13iT^{2} \)
17 \( 1 + (2.61 - 4.53i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.69 - 6.33i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-3.67 + 2.12i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.61 - 3.61i)T + 29iT^{2} \)
31 \( 1 + (-0.973 + 1.68i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (8.57 + 2.29i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 8.93iT - 41T^{2} \)
43 \( 1 + (-8.54 + 8.54i)T - 43iT^{2} \)
47 \( 1 + (-3.31 - 5.73i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.272 - 1.01i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (1.53 + 5.71i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-0.386 + 1.44i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-4.05 + 1.08i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 9.42iT - 71T^{2} \)
73 \( 1 + (5.96 + 3.44i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.83 + 3.17i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.98 - 5.98i)T + 83iT^{2} \)
89 \( 1 + (9.34 - 5.39i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 6.65T + 97T^{2} \)
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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

−10.82311137484400037733087252528, −10.20504187369230525417970812849, −9.052613176675553320959667216104, −8.378000728696717787595016816737, −7.31882567041519975674302716826, −6.68078532085899391748898728006, −5.67724665215065337005057506493, −4.52365897692956762686030378295, −3.78028553988021424775606895532, −1.87306157992547199820743293431, 0.815089824801680594459745683784, 2.66231992812927284446517113743, 3.19309752983087968456720845282, 4.78775005901364873034181542066, 5.68625998485880419873993768138, 6.40060753100015997633648390983, 8.378590650579443045298289071926, 8.882927067520105010175100633799, 9.357728087212216042125357168760, 10.74539858054605806732747917081

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