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Properties

Label 2-560-35.27-c1-0-14
Degree $2$
Conductor $560$
Sign $0.680 + 0.732i$
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  + (−0.386 − 0.386i)3-s + (0.386 + 2.20i)5-s + (−0.0564 − 2.64i)7-s − 2.70i·9-s − 1.70·11-s + (0.386 + 0.386i)13-s + (0.701 − i)15-s + (4.79 − 4.79i)17-s + 5.95·19-s + (−0.999 + 1.04i)21-s + (2.70 − 2.70i)23-s + (−4.70 + 1.70i)25-s + (−2.20 + 2.20i)27-s − 5.70i·29-s + 8.03i·31-s + ⋯
L(s)  = 1  + (−0.223 − 0.223i)3-s + (0.172 + 0.984i)5-s + (−0.0213 − 0.999i)7-s − 0.900i·9-s − 0.513·11-s + (0.107 + 0.107i)13-s + (0.181 − 0.258i)15-s + (1.16 − 1.16i)17-s + 1.36·19-s + (−0.218 + 0.227i)21-s + (0.563 − 0.563i)23-s + (−0.940 + 0.340i)25-s + (−0.423 + 0.423i)27-s − 1.05i·29-s + 1.44i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.680 + 0.732i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.680 + 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.680 + 0.732i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ 0.680 + 0.732i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24342 - 0.542112i\)
\(L(\frac12)\) \(\approx\) \(1.24342 - 0.542112i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.386 - 2.20i)T \)
7 \( 1 + (0.0564 + 2.64i)T \)
good3 \( 1 + (0.386 + 0.386i)T + 3iT^{2} \)
11 \( 1 + 1.70T + 11T^{2} \)
13 \( 1 + (-0.386 - 0.386i)T + 13iT^{2} \)
17 \( 1 + (-4.79 + 4.79i)T - 17iT^{2} \)
19 \( 1 - 5.95T + 19T^{2} \)
23 \( 1 + (-2.70 + 2.70i)T - 23iT^{2} \)
29 \( 1 + 5.70iT - 29T^{2} \)
31 \( 1 - 8.03iT - 31T^{2} \)
37 \( 1 + (-2.70 - 2.70i)T + 37iT^{2} \)
41 \( 1 + 5.95iT - 41T^{2} \)
43 \( 1 + (-5 + 5i)T - 43iT^{2} \)
47 \( 1 + (-3.24 + 3.24i)T - 47iT^{2} \)
53 \( 1 + (-5 + 5i)T - 53iT^{2} \)
59 \( 1 + 5.95T + 59T^{2} \)
61 \( 1 - 11.9iT - 61T^{2} \)
67 \( 1 + (5 + 5i)T + 67iT^{2} \)
71 \( 1 + 7.40T + 71T^{2} \)
73 \( 1 + (1.81 + 1.81i)T + 73iT^{2} \)
79 \( 1 - 0.298iT - 79T^{2} \)
83 \( 1 + (4.13 + 4.13i)T + 83iT^{2} \)
89 \( 1 - 2.08T + 89T^{2} \)
97 \( 1 + (1.15 - 1.15i)T - 97iT^{2} \)
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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

−10.50253083871589737275499459408, −10.00600492591559180141748718181, −9.047210305691908059222636193934, −7.48879023335388414958846109808, −7.24474965671652676758502097627, −6.20061393579014055944988427074, −5.16182396003916809535986799408, −3.70129014586588956908359982874, −2.86543276238106734483919515053, −0.902931082670838366749231456541, 1.50862912661249994078685309106, 2.96354749981587596369119171405, 4.47039968200936507785614265310, 5.55144829814020152292740197343, 5.74440598421613756782832334432, 7.62988356862885935609234918229, 8.158592209592975731632563756804, 9.226039023372026453809443832293, 9.862859835448333770123749829134, 10.91666668734562617484424356408

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