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Properties

Label 2-560-16.5-c1-0-3
Degree $2$
Conductor $560$
Sign $-0.576 - 0.816i$
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  + (−1.41 + 0.0784i)2-s + (0.257 − 0.257i)3-s + (1.98 − 0.221i)4-s + (−0.707 − 0.707i)5-s + (−0.343 + 0.384i)6-s + i·7-s + (−2.78 + 0.468i)8-s + 2.86i·9-s + (1.05 + 0.942i)10-s + (−1.38 − 1.38i)11-s + (0.455 − 0.569i)12-s + (−1.57 + 1.57i)13-s + (−0.0784 − 1.41i)14-s − 0.364·15-s + (3.90 − 0.881i)16-s − 5.96·17-s + ⋯
L(s)  = 1  + (−0.998 + 0.0554i)2-s + (0.148 − 0.148i)3-s + (0.993 − 0.110i)4-s + (−0.316 − 0.316i)5-s + (−0.140 + 0.156i)6-s + 0.377i·7-s + (−0.986 + 0.165i)8-s + 0.955i·9-s + (0.333 + 0.298i)10-s + (−0.416 − 0.416i)11-s + (0.131 − 0.164i)12-s + (−0.437 + 0.437i)13-s + (−0.0209 − 0.377i)14-s − 0.0941·15-s + (0.975 − 0.220i)16-s − 1.44·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.576 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.576 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $-0.576 - 0.816i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (421, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ -0.576 - 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.195580 + 0.377505i\)
\(L(\frac12)\) \(\approx\) \(0.195580 + 0.377505i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 - 0.0784i)T \)
5 \( 1 + (0.707 + 0.707i)T \)
7 \( 1 - iT \)
good3 \( 1 + (-0.257 + 0.257i)T - 3iT^{2} \)
11 \( 1 + (1.38 + 1.38i)T + 11iT^{2} \)
13 \( 1 + (1.57 - 1.57i)T - 13iT^{2} \)
17 \( 1 + 5.96T + 17T^{2} \)
19 \( 1 + (1.21 - 1.21i)T - 19iT^{2} \)
23 \( 1 - 5.13iT - 23T^{2} \)
29 \( 1 + (-2.01 + 2.01i)T - 29iT^{2} \)
31 \( 1 + 0.281T + 31T^{2} \)
37 \( 1 + (2.94 + 2.94i)T + 37iT^{2} \)
41 \( 1 - 7.72iT - 41T^{2} \)
43 \( 1 + (-7.82 - 7.82i)T + 43iT^{2} \)
47 \( 1 + 9.99T + 47T^{2} \)
53 \( 1 + (8.16 + 8.16i)T + 53iT^{2} \)
59 \( 1 + (-4.52 - 4.52i)T + 59iT^{2} \)
61 \( 1 + (6.03 - 6.03i)T - 61iT^{2} \)
67 \( 1 + (7.46 - 7.46i)T - 67iT^{2} \)
71 \( 1 - 4.27iT - 71T^{2} \)
73 \( 1 - 13.6iT - 73T^{2} \)
79 \( 1 - 11.0T + 79T^{2} \)
83 \( 1 + (-8.88 + 8.88i)T - 83iT^{2} \)
89 \( 1 + 7.06iT - 89T^{2} \)
97 \( 1 + 16.4T + 97T^{2} \)
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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

−11.14967823403114536560381908903, −10.07791594713255475049321319971, −9.198292494781733436003609662138, −8.382563436699817659637513845224, −7.77024786073282766584673099788, −6.80807937838091675020647060575, −5.72016233738473521965578702744, −4.56153223783767221161938513751, −2.85658972709216941180015719616, −1.78210080159747538506702796984, 0.30754680189565511269415735030, 2.26233305637807435733731497031, 3.40372847025708784447762253445, 4.69662802237921184004313129620, 6.34388393366967884239613212671, 6.92896754731248496709315061905, 7.87569026928785531630508861894, 8.806592376254963191013153965380, 9.496992679768150865188115333047, 10.56723895108330629721803457714

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