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Properties

Label 2-560-35.12-c1-0-1
Degree $2$
Conductor $560$
Sign $0.0542 - 0.998i$
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  + (−0.0322 + 0.120i)3-s + (−1.80 − 1.31i)5-s + (−0.969 + 2.46i)7-s + (2.58 + 1.49i)9-s + (1.40 + 2.44i)11-s + (−3.28 − 3.28i)13-s + (0.216 − 0.174i)15-s + (1.78 + 0.477i)17-s + (−4.01 + 6.95i)19-s + (−0.264 − 0.195i)21-s + (0.617 + 2.30i)23-s + (1.53 + 4.75i)25-s + (−0.526 + 0.526i)27-s + 8.63i·29-s + (2.81 − 1.62i)31-s + ⋯
L(s)  = 1  + (−0.0186 + 0.0694i)3-s + (−0.808 − 0.588i)5-s + (−0.366 + 0.930i)7-s + (0.861 + 0.497i)9-s + (0.424 + 0.736i)11-s + (−0.911 − 0.911i)13-s + (0.0559 − 0.0451i)15-s + (0.431 + 0.115i)17-s + (−0.921 + 1.59i)19-s + (−0.0577 − 0.0427i)21-s + (0.128 + 0.480i)23-s + (0.306 + 0.951i)25-s + (−0.101 + 0.101i)27-s + 1.60i·29-s + (0.505 − 0.291i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0542 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0542 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.0542 - 0.998i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ 0.0542 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.731479 + 0.692838i\)
\(L(\frac12)\) \(\approx\) \(0.731479 + 0.692838i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.80 + 1.31i)T \)
7 \( 1 + (0.969 - 2.46i)T \)
good3 \( 1 + (0.0322 - 0.120i)T + (-2.59 - 1.5i)T^{2} \)
11 \( 1 + (-1.40 - 2.44i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.28 + 3.28i)T + 13iT^{2} \)
17 \( 1 + (-1.78 - 0.477i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (4.01 - 6.95i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.617 - 2.30i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 8.63iT - 29T^{2} \)
31 \( 1 + (-2.81 + 1.62i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-6.99 + 1.87i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 9.45iT - 41T^{2} \)
43 \( 1 + (1.04 - 1.04i)T - 43iT^{2} \)
47 \( 1 + (-1.00 - 3.76i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (6.54 + 1.75i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (6.19 + 10.7i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.19 - 1.26i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.71 + 13.8i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 2.72T + 71T^{2} \)
73 \( 1 + (-1.04 + 3.90i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (5.52 + 3.18i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.41 - 7.41i)T + 83iT^{2} \)
89 \( 1 + (-0.487 + 0.844i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.12 + 5.12i)T - 97iT^{2} \)
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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

−10.96436121986295612163775440440, −9.943387099323958739933184422502, −9.366098342588299900688057690264, −8.104368852339905507762127941263, −7.68336693878749384357819461846, −6.43995118133575409367033664115, −5.24681193557147157754823426844, −4.45987507190436936800215999073, −3.25520988521568444950592012010, −1.68527224155815211560971613451, 0.59765987719875978730157009250, 2.66903666568209129059309348623, 3.98621437112017323817708568411, 4.48982379288137404474757760704, 6.38218009923773421219072425939, 6.92544024337750290351402805921, 7.63451876324951319968308392675, 8.834559111391935059030784254175, 9.778483360134978704489455740808, 10.56432659192885097933548744231

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