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Properties

Label 2-560-35.9-c1-0-14
Degree $2$
Conductor $560$
Sign $0.978 + 0.208i$
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  + (2.59 − 1.5i)3-s + (1.23 + 1.86i)5-s + (−0.866 + 2.5i)7-s + (3 − 5.19i)9-s − 2i·13-s + (6 + 3i)15-s + (1.73 − i)17-s + (1 − 1.73i)19-s + (1.5 + 7.79i)21-s + (−0.866 − 0.5i)23-s + (−1.96 + 4.59i)25-s − 9i·27-s + 29-s + (5 + 8.66i)31-s + (−5.73 + 1.46i)35-s + ⋯
L(s)  = 1  + (1.49 − 0.866i)3-s + (0.550 + 0.834i)5-s + (−0.327 + 0.944i)7-s + (1 − 1.73i)9-s − 0.554i·13-s + (1.54 + 0.774i)15-s + (0.420 − 0.242i)17-s + (0.229 − 0.397i)19-s + (0.327 + 1.70i)21-s + (−0.180 − 0.104i)23-s + (−0.392 + 0.919i)25-s − 1.73i·27-s + 0.185·29-s + (0.898 + 1.55i)31-s + (−0.968 + 0.247i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.208i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 + 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.978 + 0.208i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ 0.978 + 0.208i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.48449 - 0.262019i\)
\(L(\frac12)\) \(\approx\) \(2.48449 - 0.262019i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.23 - 1.86i)T \)
7 \( 1 + (0.866 - 2.5i)T \)
good3 \( 1 + (-2.59 + 1.5i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 + (-1.73 + i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1 + 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.866 + 0.5i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - T + 29T^{2} \)
31 \( 1 + (-5 - 8.66i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (6.92 + 4i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 3T + 41T^{2} \)
43 \( 1 + 5iT - 43T^{2} \)
47 \( 1 + (6.92 + 4i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.19 - 3i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1 + 1.73i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.5 + 7.79i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.06 - 3.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (8.66 - 5i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5 + 8.66i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 9iT - 83T^{2} \)
89 \( 1 + (3.5 - 6.06i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−10.48978388275291482992882183274, −9.670935858342770951452773601142, −8.868204735833727866287325841626, −8.162155256244728079960541215443, −7.14198250237981868038764672190, −6.49713149981109462916226624385, −5.32531175617127739089005443928, −3.35437359125743949095575594146, −2.82254329471857880622157732076, −1.77581895021649955158819762914, 1.64583999910844361337124012831, 3.06542520238267471807881358429, 4.08719133599714794544590945915, 4.78431362310224122551140672519, 6.21494961412718148333092865764, 7.57278738022660724743356351083, 8.300816762078954238820839393366, 9.111930752845880211487542675115, 9.953974472393001317832175651416, 10.16706234362097632309670879754

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