↑

Properties

Label 2-560-112.109-c1-0-13
Degree $2$
Conductor $560$
Sign $0.943 - 0.331i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.10 − 0.888i)2-s + (−2.74 + 0.735i)3-s + (0.421 + 1.95i)4-s + (0.965 + 0.258i)5-s + (3.67 + 1.62i)6-s + (−2.61 − 0.407i)7-s + (1.27 − 2.52i)8-s + (4.39 − 2.53i)9-s + (−0.832 − 1.14i)10-s + (−0.724 − 2.70i)11-s + (−2.59 − 5.05i)12-s + (−0.667 − 0.667i)13-s + (2.51 + 2.77i)14-s − 2.84·15-s + (−3.64 + 1.64i)16-s + (−2.76 + 4.78i)17-s + ⋯
L(s)  = 1  + (−0.778 − 0.628i)2-s + (−1.58 + 0.424i)3-s + (0.210 + 0.977i)4-s + (0.431 + 0.115i)5-s + (1.49 + 0.664i)6-s + (−0.988 − 0.154i)7-s + (0.449 − 0.893i)8-s + (1.46 − 0.844i)9-s + (−0.263 − 0.361i)10-s + (−0.218 − 0.815i)11-s + (−0.749 − 1.45i)12-s + (−0.185 − 0.185i)13-s + (0.671 + 0.740i)14-s − 0.733·15-s + (−0.911 + 0.412i)16-s + (−0.669 + 1.16i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 - 0.331i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.943 - 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.943 - 0.331i$
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: \((2,\ 560,\ (\ :1/2),\ 0.943 - 0.331i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.427181 + 0.0728190i\)
\(L(\frac12)\) \(\approx\) \(0.427181 + 0.0728190i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.10 + 0.888i)T \)
5 \( 1 + (-0.965 - 0.258i)T \)
7 \( 1 + (2.61 + 0.407i)T \)
good3 \( 1 + (2.74 - 0.735i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (0.724 + 2.70i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (0.667 + 0.667i)T + 13iT^{2} \)
17 \( 1 + (2.76 - 4.78i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.11 + 4.17i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-1.18 + 0.683i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5.12 - 5.12i)T + 29iT^{2} \)
31 \( 1 + (2.06 - 3.57i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.70 - 1.52i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 0.599iT - 41T^{2} \)
43 \( 1 + (-0.155 + 0.155i)T - 43iT^{2} \)
47 \( 1 + (-5.08 - 8.81i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.98 + 11.1i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-3.48 - 12.9i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-0.916 + 3.41i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-11.4 + 3.06i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 9.19iT - 71T^{2} \)
73 \( 1 + (-2.36 - 1.36i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.15 - 7.19i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-12.3 - 12.3i)T + 83iT^{2} \)
89 \( 1 + (13.5 - 7.82i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 4.69T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.87658280923850221182127448155, −10.15387114521730312264373454037, −9.401605337391826026753875048768, −8.424155143856905728502060995421, −6.90248154981498374952735855726, −6.41273508230495204471903292076, −5.32687530853682097768772616913, −4.08061371224814996544802727378, −2.83233047264089301640715404043, −0.855153281788679622713204074409, 0.56768738900091180730623748858, 2.20853286962489735903443824923, 4.62218173389736594760281739480, 5.53019614312392962929616156169, 6.27548898028665444066649985472, 6.91888688680641477133457384677, 7.72767118992193015406742175568, 9.224662388351643819737773550022, 9.840918973047741598341562258818, 10.53681245064796107694709744536

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