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Properties

Label 2-560-35.3-c1-0-8
Degree $2$
Conductor $560$
Sign $0.903 + 0.429i$
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  + (0.0200 + 0.0749i)3-s + (−2.22 − 0.172i)5-s + (−2.20 + 1.45i)7-s + (2.59 − 1.49i)9-s + (0.390 − 0.677i)11-s + (3.28 − 3.28i)13-s + (−0.0318 − 0.170i)15-s + (5.24 − 1.40i)17-s + (2.91 + 5.05i)19-s + (−0.153 − 0.136i)21-s + (2.24 − 8.39i)23-s + (4.94 + 0.770i)25-s + (0.328 + 0.328i)27-s + 0.303i·29-s + (5.04 + 2.91i)31-s + ⋯
L(s)  = 1  + (0.0115 + 0.0432i)3-s + (−0.997 − 0.0772i)5-s + (−0.834 + 0.551i)7-s + (0.864 − 0.498i)9-s + (0.117 − 0.204i)11-s + (0.911 − 0.911i)13-s + (−0.00821 − 0.0440i)15-s + (1.27 − 0.341i)17-s + (0.669 + 1.15i)19-s + (−0.0335 − 0.0297i)21-s + (0.468 − 1.74i)23-s + (0.988 + 0.154i)25-s + (0.0632 + 0.0632i)27-s + 0.0564i·29-s + (0.905 + 0.522i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.903 + 0.429i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.903 + 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.903 + 0.429i$
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: \((2,\ 560,\ (\ :1/2),\ 0.903 + 0.429i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22156 - 0.275560i\)
\(L(\frac12)\) \(\approx\) \(1.22156 - 0.275560i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.22 + 0.172i)T \)
7 \( 1 + (2.20 - 1.45i)T \)
good3 \( 1 + (-0.0200 - 0.0749i)T + (-2.59 + 1.5i)T^{2} \)
11 \( 1 + (-0.390 + 0.677i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.28 + 3.28i)T - 13iT^{2} \)
17 \( 1 + (-5.24 + 1.40i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-2.91 - 5.05i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.24 + 8.39i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 0.303iT - 29T^{2} \)
31 \( 1 + (-5.04 - 2.91i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (7.12 + 1.90i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 4.39iT - 41T^{2} \)
43 \( 1 + (7.33 + 7.33i)T + 43iT^{2} \)
47 \( 1 + (0.611 - 2.28i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-6.01 + 1.61i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-2.56 + 4.43i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (8.91 - 5.14i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.395 + 1.47i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 5.83T + 71T^{2} \)
73 \( 1 + (-3.47 - 12.9i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-8.66 + 5.00i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.94 + 2.94i)T - 83iT^{2} \)
89 \( 1 + (6.43 + 11.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.0900 - 0.0900i)T + 97iT^{2} \)
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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

−10.49774875342831403473042355310, −9.996293130287655739502744411971, −8.804540547388645115892605149702, −8.140229726467233119695767740094, −7.09297493403709855797839177578, −6.20824227582132783467987201741, −5.10390939862882690503082512941, −3.71709145044279097190926974605, −3.15838679355762291482414999089, −0.921590351478755637259053357228, 1.24964095562831465847998044961, 3.26764484559070111706309737296, 3.98702660746389322773984557615, 5.09724302203835243331742491195, 6.56902303561526954291448647423, 7.24177258182050972352652825028, 7.962559061306389039491994237429, 9.180472062639277960379946573360, 9.939747167964013863943883177092, 10.86658034844952845454744768976

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