Properties

Label 2-560-35.33-c1-0-9
Degree $2$
Conductor $560$
Sign $0.490 - 0.871i$
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  + (2.80 + 0.752i)3-s + (−2.21 − 0.318i)5-s + (−0.559 + 2.58i)7-s + (4.71 + 2.72i)9-s + (1.83 + 3.17i)11-s + (0.830 − 0.830i)13-s + (−5.97 − 2.55i)15-s + (−0.204 + 0.761i)17-s + (−1.09 + 1.89i)19-s + (−3.51 + 6.83i)21-s + (4.54 − 1.21i)23-s + (4.79 + 1.41i)25-s + (5.03 + 5.03i)27-s − 2.62i·29-s + (−0.0359 + 0.0207i)31-s + ⋯
L(s)  = 1  + (1.62 + 0.434i)3-s + (−0.989 − 0.142i)5-s + (−0.211 + 0.977i)7-s + (1.57 + 0.908i)9-s + (0.553 + 0.958i)11-s + (0.230 − 0.230i)13-s + (−1.54 − 0.660i)15-s + (−0.0494 + 0.184i)17-s + (−0.251 + 0.434i)19-s + (−0.767 + 1.49i)21-s + (0.947 − 0.253i)23-s + (0.959 + 0.282i)25-s + (0.968 + 0.968i)27-s − 0.486i·29-s + (−0.00645 + 0.00372i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.490 - 0.871i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.490 - 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.490 - 0.871i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ 0.490 - 0.871i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.82458 + 1.06637i\)
\(L(\frac12)\) \(\approx\) \(1.82458 + 1.06637i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.21 + 0.318i)T \)
7 \( 1 + (0.559 - 2.58i)T \)
good3 \( 1 + (-2.80 - 0.752i)T + (2.59 + 1.5i)T^{2} \)
11 \( 1 + (-1.83 - 3.17i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.830 + 0.830i)T - 13iT^{2} \)
17 \( 1 + (0.204 - 0.761i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.09 - 1.89i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.54 + 1.21i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 2.62iT - 29T^{2} \)
31 \( 1 + (0.0359 - 0.0207i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.0664 - 0.248i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 8.98iT - 41T^{2} \)
43 \( 1 + (-0.474 - 0.474i)T + 43iT^{2} \)
47 \( 1 + (6.18 - 1.65i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-2.04 + 7.64i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (5.35 + 9.27i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.72 - 0.996i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.39 + 1.71i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 8.11T + 71T^{2} \)
73 \( 1 + (-9.52 - 2.55i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-11.6 - 6.70i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-9.73 + 9.73i)T - 83iT^{2} \)
89 \( 1 + (-0.715 + 1.23i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.16 + 3.16i)T + 97iT^{2} \)
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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

−10.77735456788294631463081444344, −9.701914741639628526786265938990, −9.048598313785667183745639194230, −8.385380492313742612034171977459, −7.66960496085604633073140305885, −6.62451409751635347555271199024, −4.99416335550864480966049200197, −3.99600874712116489895107547071, −3.18112761388463377593801172958, −2.02580172678822115162962939002, 1.14430807006125117212215313834, 2.96226160292868466738086971140, 3.57912655116221750734908687567, 4.51870565034308881986408638437, 6.51704484056114009606622405572, 7.24009385347305896535460607268, 7.970001537866975627358695893485, 8.748743149252052224781833392684, 9.406074905954169337886532857228, 10.67062288733177518175564463151

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