Properties

Label 2-560-35.17-c1-0-11
Degree $2$
Conductor $560$
Sign $0.945 - 0.325i$
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  + (2.00 − 0.536i)3-s + (1.38 + 1.75i)5-s + (−1.24 + 2.33i)7-s + (1.11 − 0.644i)9-s + (3.09 − 5.36i)11-s + (0.782 + 0.782i)13-s + (3.70 + 2.77i)15-s + (1.18 + 4.40i)17-s + (−2.37 − 4.11i)19-s + (−1.25 + 5.33i)21-s + (6.52 + 1.74i)23-s + (−1.17 + 4.86i)25-s + (−2.50 + 2.50i)27-s + 5.30i·29-s + (−2.16 − 1.25i)31-s + ⋯
L(s)  = 1  + (1.15 − 0.309i)3-s + (0.618 + 0.785i)5-s + (−0.472 + 0.881i)7-s + (0.372 − 0.214i)9-s + (0.933 − 1.61i)11-s + (0.217 + 0.217i)13-s + (0.957 + 0.715i)15-s + (0.286 + 1.06i)17-s + (−0.544 − 0.943i)19-s + (−0.272 + 1.16i)21-s + (1.35 + 0.364i)23-s + (−0.234 + 0.972i)25-s + (−0.482 + 0.482i)27-s + 0.985i·29-s + (−0.389 − 0.224i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 - 0.325i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.945 - 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.945 - 0.325i$
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: \((2,\ 560,\ (\ :1/2),\ 0.945 - 0.325i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.24299 + 0.375807i\)
\(L(\frac12)\) \(\approx\) \(2.24299 + 0.375807i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.38 - 1.75i)T \)
7 \( 1 + (1.24 - 2.33i)T \)
good3 \( 1 + (-2.00 + 0.536i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (-3.09 + 5.36i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.782 - 0.782i)T + 13iT^{2} \)
17 \( 1 + (-1.18 - 4.40i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (2.37 + 4.11i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6.52 - 1.74i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 - 5.30iT - 29T^{2} \)
31 \( 1 + (2.16 + 1.25i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.77 + 6.61i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 2.51iT - 41T^{2} \)
43 \( 1 + (0.404 - 0.404i)T - 43iT^{2} \)
47 \( 1 + (8.63 + 2.31i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.66 + 9.92i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (0.0710 - 0.123i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.67 + 1.54i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.34 + 1.16i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 10.3T + 71T^{2} \)
73 \( 1 + (1.80 - 0.483i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (5.42 - 3.13i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.82 + 5.82i)T + 83iT^{2} \)
89 \( 1 + (4.58 + 7.94i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.64 + 2.64i)T - 97iT^{2} \)
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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

−10.95120102285666890919017159781, −9.651153698579406537103964952393, −8.802502610845766261532179525106, −8.589134722966036127954345790081, −7.14718420300122612024866557902, −6.32806177563389483048558576883, −5.52982116181684819806000541386, −3.53243678774996314067260107136, −3.01086380720290778384208639209, −1.81222544672207163419311927045, 1.42546711650034428755199152792, 2.81827320996772517760360804947, 4.05106943782280050147198686111, 4.77739069565901949712955193890, 6.27442438366804474414002831061, 7.24247346966831244265525611092, 8.188936147330630550765979323217, 9.188258182566900250184018980178, 9.662030753433901063448467400034, 10.25098172938199230604421436957

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