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Properties

Label 2-560-35.33-c1-0-8
Degree $2$
Conductor $560$
Sign $0.284 + 0.958i$
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  + (−2.19 − 0.588i)3-s + (−1.19 + 1.88i)5-s + (−1.40 + 2.24i)7-s + (1.88 + 1.08i)9-s + (−1.78 − 3.09i)11-s + (3.13 − 3.13i)13-s + (3.74 − 3.44i)15-s + (−1.34 + 5.00i)17-s + (0.687 − 1.19i)19-s + (4.40 − 4.10i)21-s + (4.00 − 1.07i)23-s + (−2.12 − 4.52i)25-s + (1.32 + 1.32i)27-s − 9.39i·29-s + (6.08 − 3.51i)31-s + ⋯
L(s)  = 1  + (−1.26 − 0.340i)3-s + (−0.536 + 0.844i)5-s + (−0.530 + 0.847i)7-s + (0.628 + 0.363i)9-s + (−0.539 − 0.934i)11-s + (0.869 − 0.869i)13-s + (0.967 − 0.888i)15-s + (−0.325 + 1.21i)17-s + (0.157 − 0.273i)19-s + (0.961 − 0.895i)21-s + (0.835 − 0.223i)23-s + (−0.425 − 0.905i)25-s + (0.254 + 0.254i)27-s − 1.74i·29-s + (1.09 − 0.630i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.284 + 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.284 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.284 + 0.958i$
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: \((2,\ 560,\ (\ :1/2),\ 0.284 + 0.958i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.423643 - 0.316073i\)
\(L(\frac12)\) \(\approx\) \(0.423643 - 0.316073i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.19 - 1.88i)T \)
7 \( 1 + (1.40 - 2.24i)T \)
good3 \( 1 + (2.19 + 0.588i)T + (2.59 + 1.5i)T^{2} \)
11 \( 1 + (1.78 + 3.09i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.13 + 3.13i)T - 13iT^{2} \)
17 \( 1 + (1.34 - 5.00i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-0.687 + 1.19i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.00 + 1.07i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 9.39iT - 29T^{2} \)
31 \( 1 + (-6.08 + 3.51i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.68 + 6.29i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 4.63iT - 41T^{2} \)
43 \( 1 + (2.51 + 2.51i)T + 43iT^{2} \)
47 \( 1 + (8.76 - 2.34i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (0.470 - 1.75i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-1.42 - 2.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.91 - 1.10i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.37 - 1.17i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 10.5T + 71T^{2} \)
73 \( 1 + (-2.44 - 0.654i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-4.82 - 2.78i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.863 - 0.863i)T - 83iT^{2} \)
89 \( 1 + (-0.430 + 0.745i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-12.6 - 12.6i)T + 97iT^{2} \)
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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

−10.91476427944761191022729698708, −10.04782060405159096272159384979, −8.626493164947481854920430811014, −7.910755091650365972758824121974, −6.56563665687431937722611262611, −6.11296869411454291308792127112, −5.36504642601035850188684691371, −3.76082134785032226651593013258, −2.65559461432403958510061052175, −0.42603265373705332806319402811, 1.13325594048886686456246374723, 3.43784095280356186057154741165, 4.84835658646641286815865604528, 4.89338613504621475407752539158, 6.50111255929813276633111160758, 7.08573274461463714507932259541, 8.302348340841981430257811029743, 9.394074832656801078983758638396, 10.14859160734470117125289110738, 11.10154065265199089745753487370

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