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Properties

Label 2-560-112.109-c1-0-33
Degree $2$
Conductor $560$
Sign $0.862 + 0.505i$
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  + (−1.31 + 0.511i)2-s + (−0.0714 + 0.0191i)3-s + (1.47 − 1.34i)4-s + (−0.965 − 0.258i)5-s + (0.0843 − 0.0617i)6-s + (1.28 − 2.31i)7-s + (−1.25 + 2.53i)8-s + (−2.59 + 1.49i)9-s + (1.40 − 0.153i)10-s + (0.200 + 0.749i)11-s + (−0.0795 + 0.124i)12-s + (3.55 + 3.55i)13-s + (−0.513 + 3.70i)14-s + 0.0739·15-s + (0.355 − 3.98i)16-s + (2.94 − 5.09i)17-s + ⋯
L(s)  = 1  + (−0.932 + 0.362i)2-s + (−0.0412 + 0.0110i)3-s + (0.737 − 0.674i)4-s + (−0.431 − 0.115i)5-s + (0.0344 − 0.0252i)6-s + (0.486 − 0.873i)7-s + (−0.443 + 0.896i)8-s + (−0.864 + 0.499i)9-s + (0.444 − 0.0484i)10-s + (0.0605 + 0.225i)11-s + (−0.0229 + 0.0359i)12-s + (0.986 + 0.986i)13-s + (−0.137 + 0.990i)14-s + 0.0190·15-s + (0.0889 − 0.996i)16-s + (0.713 − 1.23i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.862 + 0.505i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.862 + 0.505i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.862 + 0.505i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ 0.862 + 0.505i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.827294 - 0.224379i\)
\(L(\frac12)\) \(\approx\) \(0.827294 - 0.224379i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.31 - 0.511i)T \)
5 \( 1 + (0.965 + 0.258i)T \)
7 \( 1 + (-1.28 + 2.31i)T \)
good3 \( 1 + (0.0714 - 0.0191i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (-0.200 - 0.749i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (-3.55 - 3.55i)T + 13iT^{2} \)
17 \( 1 + (-2.94 + 5.09i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.576 + 2.15i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.800 + 0.462i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (6.32 + 6.32i)T + 29iT^{2} \)
31 \( 1 + (-3.30 + 5.72i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-7.46 - 2.00i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 4.60iT - 41T^{2} \)
43 \( 1 + (-7.39 + 7.39i)T - 43iT^{2} \)
47 \( 1 + (-3.78 - 6.56i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.34 + 5.03i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-1.91 - 7.14i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (1.04 - 3.89i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-15.2 + 4.07i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 7.90iT - 71T^{2} \)
73 \( 1 + (12.9 + 7.45i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.96 - 8.59i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.47 - 4.47i)T + 83iT^{2} \)
89 \( 1 + (4.74 - 2.74i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 2.86T + 97T^{2} \)
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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

−10.78826676416605809317015931767, −9.628004277015423322675605791054, −8.932239158076487375041336064350, −7.87928047746246924030865090202, −7.44200560540096274727608587218, −6.32241499053523703471898930862, −5.25409675041287332628206720790, −4.06579453781076657542901405741, −2.41279685554306509648538800119, −0.78638912930444979477199941576, 1.26364203362774226277238375343, 2.90155237757487087017621956065, 3.70114857223904882502317357709, 5.60172662689727646516687186612, 6.24304231137078576211957509681, 7.67081719453639936898538836580, 8.383246715269803202758485901292, 8.846989750310425358989565147367, 9.975569551668870294151309755185, 11.04298624043927858930025833927

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