Properties

Label 2-560-35.3-c1-0-10
Degree $2$
Conductor $560$
Sign $0.920 - 0.391i$
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  + (0.0749 + 0.279i)3-s + (2.20 + 0.378i)5-s + (−0.126 + 2.64i)7-s + (2.52 − 1.45i)9-s + (2.81 − 4.87i)11-s + (−1.42 + 1.42i)13-s + (0.0593 + 0.645i)15-s + (−5.12 + 1.37i)17-s + (1.94 + 3.37i)19-s + (−0.749 + 0.162i)21-s + (0.290 − 1.08i)23-s + (4.71 + 1.66i)25-s + (1.21 + 1.21i)27-s + 3.15i·29-s + (3.33 + 1.92i)31-s + ⋯
L(s)  = 1  + (0.0432 + 0.161i)3-s + (0.985 + 0.169i)5-s + (−0.0477 + 0.998i)7-s + (0.841 − 0.486i)9-s + (0.848 − 1.46i)11-s + (−0.396 + 0.396i)13-s + (0.0153 + 0.166i)15-s + (−1.24 + 0.333i)17-s + (0.446 + 0.773i)19-s + (−0.163 + 0.0355i)21-s + (0.0606 − 0.226i)23-s + (0.942 + 0.333i)25-s + (0.233 + 0.233i)27-s + 0.585i·29-s + (0.598 + 0.345i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.391i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.920 - 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.920 - 0.391i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ 0.920 - 0.391i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.81231 + 0.369788i\)
\(L(\frac12)\) \(\approx\) \(1.81231 + 0.369788i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.20 - 0.378i)T \)
7 \( 1 + (0.126 - 2.64i)T \)
good3 \( 1 + (-0.0749 - 0.279i)T + (-2.59 + 1.5i)T^{2} \)
11 \( 1 + (-2.81 + 4.87i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.42 - 1.42i)T - 13iT^{2} \)
17 \( 1 + (5.12 - 1.37i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.94 - 3.37i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.290 + 1.08i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 3.15iT - 29T^{2} \)
31 \( 1 + (-3.33 - 1.92i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.86 - 1.30i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 7.21iT - 41T^{2} \)
43 \( 1 + (1.85 + 1.85i)T + 43iT^{2} \)
47 \( 1 + (-1.52 + 5.69i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-1.33 + 0.357i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (2.73 - 4.74i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.99 - 2.30i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.218 + 0.816i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 4.77T + 71T^{2} \)
73 \( 1 + (-1.45 - 5.42i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (5.41 - 3.12i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.67 - 5.67i)T - 83iT^{2} \)
89 \( 1 + (5.96 + 10.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.63 + 6.63i)T + 97iT^{2} \)
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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

−10.74632408758606582001330876519, −9.835217709329107180600240281292, −9.037428455169884523321055584514, −8.564380444747450089352885718787, −6.92141570933667041947509934004, −6.25047574489053493224125966521, −5.44151527465392646410883779578, −4.11293586213264407253774881141, −2.87150920369792463750134790694, −1.55060934657648868854110206294, 1.34378497335412974290400247376, 2.49953350242194448395823413790, 4.36800858454957860020588659134, 4.80115572274141205991796908796, 6.40212649448309153552068921735, 7.04168828797342799948969825587, 7.82406154520146323433657873285, 9.343800511787382606349138634305, 9.720448180482578147170197124718, 10.52595736143494319862056570040

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