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Properties

Label 2-560-140.67-c1-0-12
Degree $2$
Conductor $560$
Sign $-0.164 + 0.986i$
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  + (−1.67 + 0.448i)3-s + (−1.23 + 1.86i)5-s + (1.48 − 2.19i)7-s + (−2.12 − 1.22i)11-s + (−2 + 2i)13-s + (1.22 − 3.67i)15-s + (2.73 − 0.732i)17-s + (−1.22 − 2.12i)19-s + (−1.50 + 4.33i)21-s + (1.34 − 5.01i)23-s + (−1.96 − 4.59i)25-s + (3.67 − 3.67i)27-s − i·29-s + (−4.24 − 2.44i)31-s + (4.09 + 1.09i)33-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)3-s + (−0.550 + 0.834i)5-s + (0.560 − 0.827i)7-s + (−0.639 − 0.369i)11-s + (−0.554 + 0.554i)13-s + (0.316 − 0.948i)15-s + (0.662 − 0.177i)17-s + (−0.280 − 0.486i)19-s + (−0.327 + 0.944i)21-s + (0.280 − 1.04i)23-s + (−0.392 − 0.919i)25-s + (0.707 − 0.707i)27-s − 0.185i·29-s + (−0.762 − 0.439i)31-s + (0.713 + 0.191i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.164 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.164 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $-0.164 + 0.986i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (207, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ -0.164 + 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.284167 - 0.335406i\)
\(L(\frac12)\) \(\approx\) \(0.284167 - 0.335406i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.48 + 2.19i)T \)
good3 \( 1 + (1.67 - 0.448i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (2.12 + 1.22i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2 - 2i)T - 13iT^{2} \)
17 \( 1 + (-2.73 + 0.732i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.22 + 2.12i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.34 + 5.01i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + iT - 29T^{2} \)
31 \( 1 + (4.24 + 2.44i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.92 + 10.9i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + T + 41T^{2} \)
43 \( 1 + (6.12 + 6.12i)T + 43iT^{2} \)
47 \( 1 + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (0.366 + 1.36i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (6.12 - 10.6i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.5 - 2.59i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.448 - 1.67i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 12.2iT - 71T^{2} \)
73 \( 1 + (-0.732 - 2.73i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (8.57 + 14.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.67 - 3.67i)T + 83iT^{2} \)
89 \( 1 + (9.52 - 5.5i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-11 - 11i)T + 97iT^{2} \)
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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

−10.65865567210577013178926918945, −10.10166963919708747798597874431, −8.661636027934014846507007109380, −7.61350925262893894748438068810, −6.98551707154751367035378673870, −5.87668127658269609356526133354, −4.87679727874017594130544567587, −3.96622674092309414154491441673, −2.53965562347722063144209943465, −0.29107762368289968905549254727, 1.47406202658271164358034222021, 3.22619547803167121673982748360, 4.93314509385603071110532942910, 5.24045855770353699272921779856, 6.25778252653800107182209317899, 7.61834853870185504188004225923, 8.187316748045202053931810419173, 9.217160827424358861855678198765, 10.22570583458655467713893654389, 11.31538470565394298507553408687

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