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Properties

Label 2-560-35.3-c1-0-12
Degree $2$
Conductor $560$
Sign $0.873 - 0.486i$
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  + (0.536 + 2.00i)3-s + (−0.829 − 2.07i)5-s + (2.33 + 1.24i)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 + (4.40 − 1.18i)17-s + (2.37 + 4.11i)19-s + (−1.25 + 5.33i)21-s + (−1.74 + 6.52i)23-s + (−3.62 + 3.44i)25-s + (2.50 + 2.50i)27-s − 5.30i·29-s + (−2.16 − 1.25i)31-s + ⋯
L(s)  = 1  + (0.309 + 1.15i)3-s + (−0.370 − 0.928i)5-s + (0.881 + 0.472i)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 + (1.06 − 0.286i)17-s + (0.544 + 0.943i)19-s + (−0.272 + 1.16i)21-s + (−0.364 + 1.35i)23-s + (−0.724 + 0.688i)25-s + (0.482 + 0.482i)27-s − 0.985i·29-s + (−0.389 − 0.224i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.873 - 0.486i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.873 - 0.486i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.873 - 0.486i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ 0.873 - 0.486i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.71775 + 0.446180i\)
\(L(\frac12)\) \(\approx\) \(1.71775 + 0.446180i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.829 + 2.07i)T \)
7 \( 1 + (-2.33 - 1.24i)T \)
good3 \( 1 + (-0.536 - 2.00i)T + (-2.59 + 1.5i)T^{2} \)
11 \( 1 + (-3.09 + 5.36i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.782 - 0.782i)T - 13iT^{2} \)
17 \( 1 + (-4.40 + 1.18i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-2.37 - 4.11i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.74 - 6.52i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + 5.30iT - 29T^{2} \)
31 \( 1 + (2.16 + 1.25i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (6.61 + 1.77i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 2.51iT - 41T^{2} \)
43 \( 1 + (0.404 + 0.404i)T + 43iT^{2} \)
47 \( 1 + (2.31 - 8.63i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-9.92 + 2.66i)T + (45.8 - 26.5i)T^{2} \)
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 \( 1 + (1.16 + 4.34i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 10.3T + 71T^{2} \)
73 \( 1 + (0.483 + 1.80i)T + (-63.2 + 36.5i)T^{2} \)
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

−10.86121192555439435957577567431, −9.700936754120911305246096481465, −9.146306978541541908540587302986, −8.375921509533373825866120975365, −7.62809958243476487682398040893, −5.80790497065233482564945851125, −5.23684567291595485972643071038, −4.04211053016959376635178739443, −3.42780210161271641517988626717, −1.35334498383104048206923498996, 1.38956076933278703690002827672, 2.47748119005280807533440156492, 3.91132885465822627450486847574, 5.03062567316781707127447048849, 6.69023166757876661978143979665, 7.10607689782822358225503319689, 7.72418238247953163344144058515, 8.670477850360939438144895444755, 10.02343728154995609187592731712, 10.62945439387226354749554486898

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