Properties

Label 2-560-35.33-c1-0-17
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: $\chi_{560} (33, \cdot )$
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

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

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