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Properties

Label 2-560-35.12-c1-0-10
Degree $2$
Conductor $560$
Sign $0.973 + 0.226i$
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  + (0.312 − 1.16i)3-s + (1.87 − 1.22i)5-s + (−0.491 + 2.59i)7-s + (1.33 + 0.770i)9-s + (1.67 + 2.90i)11-s + (2.92 + 2.92i)13-s + (−0.840 − 2.56i)15-s + (0.259 + 0.0694i)17-s + (−0.458 + 0.793i)19-s + (2.88 + 1.38i)21-s + (−2.03 − 7.58i)23-s + (2.01 − 4.57i)25-s + (3.87 − 3.87i)27-s − 1.31i·29-s + (−3.25 + 1.88i)31-s + ⋯
L(s)  = 1  + (0.180 − 0.673i)3-s + (0.837 − 0.546i)5-s + (−0.185 + 0.982i)7-s + (0.444 + 0.256i)9-s + (0.506 + 0.877i)11-s + (0.810 + 0.810i)13-s + (−0.216 − 0.662i)15-s + (0.0628 + 0.0168i)17-s + (−0.105 + 0.182i)19-s + (0.628 + 0.302i)21-s + (−0.423 − 1.58i)23-s + (0.402 − 0.915i)25-s + (0.746 − 0.746i)27-s − 0.244i·29-s + (−0.584 + 0.337i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.226i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 + 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.973 + 0.226i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ 0.973 + 0.226i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.87252 - 0.215307i\)
\(L(\frac12)\) \(\approx\) \(1.87252 - 0.215307i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.87 + 1.22i)T \)
7 \( 1 + (0.491 - 2.59i)T \)
good3 \( 1 + (-0.312 + 1.16i)T + (-2.59 - 1.5i)T^{2} \)
11 \( 1 + (-1.67 - 2.90i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.92 - 2.92i)T + 13iT^{2} \)
17 \( 1 + (-0.259 - 0.0694i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (0.458 - 0.793i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.03 + 7.58i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 1.31iT - 29T^{2} \)
31 \( 1 + (3.25 - 1.88i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (7.90 - 2.11i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 5.65iT - 41T^{2} \)
43 \( 1 + (-2.61 + 2.61i)T - 43iT^{2} \)
47 \( 1 + (-0.244 - 0.911i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (13.0 + 3.49i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-3.91 - 6.78i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-9.70 - 5.60i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.44 + 5.37i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 13.5T + 71T^{2} \)
73 \( 1 + (1.96 - 7.33i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (1.87 + 1.08i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.49 + 5.49i)T + 83iT^{2} \)
89 \( 1 + (-1.34 + 2.33i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-10.1 + 10.1i)T - 97iT^{2} \)
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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

−10.60212914194997575714974389513, −9.728095004917533290454616318308, −8.900432305791748646523706507936, −8.284974669965505686757474132964, −6.89792253921030511185465364451, −6.34374193723736455143331636910, −5.22545176010823718640123219380, −4.15156854585769926542095073200, −2.32473860030277334896931208642, −1.59789310313471818212477723005, 1.34578709386512204618493698698, 3.29954190738444181788032679328, 3.79462050783931342352237265739, 5.26941052046303454113170632255, 6.24509768428667730304505981641, 7.08290302190529803194875329402, 8.199474341498099554274434135299, 9.397123002993777456853929626864, 9.824997573518388723328272524218, 10.81572153860358519662817378738

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