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Properties

Label 2-560-35.17-c1-0-19
Degree $2$
Conductor $560$
Sign $0.698 + 0.715i$
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  + (2.30 − 0.617i)3-s + (1.37 − 1.76i)5-s + (0.755 − 2.53i)7-s + (2.33 − 1.34i)9-s + (−2.18 + 3.78i)11-s + (4.36 + 4.36i)13-s + (2.09 − 4.91i)15-s + (0.438 + 1.63i)17-s + (−3.56 − 6.17i)19-s + (0.175 − 6.31i)21-s + (−4.91 − 1.31i)23-s + (−1.19 − 4.85i)25-s + (−0.510 + 0.510i)27-s − 1.33i·29-s + (1.90 + 1.09i)31-s + ⋯
L(s)  = 1  + (1.33 − 0.356i)3-s + (0.616 − 0.787i)5-s + (0.285 − 0.958i)7-s + (0.778 − 0.449i)9-s + (−0.659 + 1.14i)11-s + (1.21 + 1.21i)13-s + (0.539 − 1.26i)15-s + (0.106 + 0.396i)17-s + (−0.818 − 1.41i)19-s + (0.0382 − 1.37i)21-s + (−1.02 − 0.274i)23-s + (−0.239 − 0.970i)25-s + (−0.0982 + 0.0982i)27-s − 0.247i·29-s + (0.341 + 0.197i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.698 + 0.715i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.698 + 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.698 + 0.715i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ 0.698 + 0.715i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.25499 - 0.949570i\)
\(L(\frac12)\) \(\approx\) \(2.25499 - 0.949570i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.37 + 1.76i)T \)
7 \( 1 + (-0.755 + 2.53i)T \)
good3 \( 1 + (-2.30 + 0.617i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (2.18 - 3.78i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.36 - 4.36i)T + 13iT^{2} \)
17 \( 1 + (-0.438 - 1.63i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (3.56 + 6.17i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.91 + 1.31i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 1.33iT - 29T^{2} \)
31 \( 1 + (-1.90 - 1.09i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.224 - 0.839i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 5.69iT - 41T^{2} \)
43 \( 1 + (3.40 - 3.40i)T - 43iT^{2} \)
47 \( 1 + (-9.84 - 2.63i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (0.541 + 2.02i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (2.56 - 4.44i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.21 + 3.58i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.11 + 1.90i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 4.31T + 71T^{2} \)
73 \( 1 + (14.2 - 3.82i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (4.82 - 2.78i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.272 - 0.272i)T + 83iT^{2} \)
89 \( 1 + (-1.79 - 3.10i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.325 + 0.325i)T - 97iT^{2} \)
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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

−10.42330477796448691297066213309, −9.610114160592783849430662028573, −8.761788659811548646176349913409, −8.184734342336745606770946107475, −7.21931540330467297599001771008, −6.29326069910183048300819033143, −4.69075311930587602806767504960, −4.03258673693708398027315901992, −2.40862591902852225703932727856, −1.51271879678408269986129442484, 2.06506997902518269915060785743, 3.02638869000414173757437645984, 3.72624192151165113468029803750, 5.66236442131430751937111898795, 5.98334385946414782544554791933, 7.68043566840866706171186260696, 8.404081148266674294547820589800, 8.870931647029508424716075815784, 10.09148368063447640483385800053, 10.55247710238159451408767025723

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