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Properties

Label 2-560-7.4-c1-0-3
Degree $2$
Conductor $560$
Sign $0.386 - 0.922i$
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  + (0.5 − 0.866i)3-s + (0.5 + 0.866i)5-s + (0.5 + 2.59i)7-s + (1 + 1.73i)9-s + (−3 + 5.19i)11-s − 4·13-s + 0.999·15-s + (1 + 1.73i)19-s + (2.5 + 0.866i)21-s + (−1.5 − 2.59i)23-s + (−0.499 + 0.866i)25-s + 5·27-s − 3·29-s + (4 − 6.92i)31-s + (3 + 5.19i)33-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (0.223 + 0.387i)5-s + (0.188 + 0.981i)7-s + (0.333 + 0.577i)9-s + (−0.904 + 1.56i)11-s − 1.10·13-s + 0.258·15-s + (0.229 + 0.397i)19-s + (0.545 + 0.188i)21-s + (−0.312 − 0.541i)23-s + (−0.0999 + 0.173i)25-s + 0.962·27-s − 0.557·29-s + (0.718 − 1.24i)31-s + (0.522 + 0.904i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.386 - 0.922i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ 0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21848 + 0.810515i\)
\(L(\frac12)\) \(\approx\) \(1.21848 + 0.810515i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-0.5 - 2.59i)T \)
good3 \( 1 + (-0.5 + 0.866i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (3 - 5.19i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1 - 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 + (-4 + 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2 - 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 9T + 41T^{2} \)
43 \( 1 - 7T + 43T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3 - 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.5 + 4.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.5 + 4.33i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + (-8 + 13.8i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1 - 1.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3T + 83T^{2} \)
89 \( 1 + (-7.5 - 12.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 14T + 97T^{2} \)
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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

−10.82465588357168608819917333454, −9.950990715049745388256337962014, −9.345121989817046898139113135481, −7.84031517987816893486907137675, −7.66773938004256374424419296030, −6.49011580126593528066955492493, −5.30905693630335057621821325933, −4.49408495836826359745648464830, −2.51464033325386093398727874030, −2.13931140405510732205263691279, 0.811805894555582142530315413422, 2.80063679728376044789761690869, 3.88345065625122588213397001023, 4.88254824613117505382671804894, 5.88088910561294881195022216836, 7.13977471800601991289190397013, 7.958895390944305290544764811325, 8.954531385528270619857509295449, 9.737916287566358540414165534832, 10.52610291090482988796514097006

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