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Properties

Label 2-560-16.5-c1-0-12
Degree $2$
Conductor $560$
Sign $-0.189 - 0.981i$
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  + (−1.04 + 0.948i)2-s + (0.448 − 0.448i)3-s + (0.201 − 1.98i)4-s + (0.707 + 0.707i)5-s + (−0.0452 + 0.895i)6-s + i·7-s + (1.67 + 2.27i)8-s + 2.59i·9-s + (−1.41 − 0.0713i)10-s + (−1.28 − 1.28i)11-s + (−0.802 − 0.982i)12-s + (−2.40 + 2.40i)13-s + (−0.948 − 1.04i)14-s + 0.634·15-s + (−3.91 − 0.802i)16-s + 5.53·17-s + ⋯
L(s)  = 1  + (−0.741 + 0.670i)2-s + (0.258 − 0.258i)3-s + (0.100 − 0.994i)4-s + (0.316 + 0.316i)5-s + (−0.0184 + 0.365i)6-s + 0.377i·7-s + (0.592 + 0.805i)8-s + 0.865i·9-s + (−0.446 − 0.0225i)10-s + (−0.387 − 0.387i)11-s + (−0.231 − 0.283i)12-s + (−0.666 + 0.666i)13-s + (−0.253 − 0.280i)14-s + 0.163·15-s + (−0.979 − 0.200i)16-s + 1.34·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.189 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.189 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $-0.189 - 0.981i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (421, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ -0.189 - 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.647298 + 0.784290i\)
\(L(\frac12)\) \(\approx\) \(0.647298 + 0.784290i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.04 - 0.948i)T \)
5 \( 1 + (-0.707 - 0.707i)T \)
7 \( 1 - iT \)
good3 \( 1 + (-0.448 + 0.448i)T - 3iT^{2} \)
11 \( 1 + (1.28 + 1.28i)T + 11iT^{2} \)
13 \( 1 + (2.40 - 2.40i)T - 13iT^{2} \)
17 \( 1 - 5.53T + 17T^{2} \)
19 \( 1 + (1.67 - 1.67i)T - 19iT^{2} \)
23 \( 1 - 0.722iT - 23T^{2} \)
29 \( 1 + (3.78 - 3.78i)T - 29iT^{2} \)
31 \( 1 - 3.07T + 31T^{2} \)
37 \( 1 + (-6.66 - 6.66i)T + 37iT^{2} \)
41 \( 1 - 1.68iT - 41T^{2} \)
43 \( 1 + (-4.96 - 4.96i)T + 43iT^{2} \)
47 \( 1 + 2.72T + 47T^{2} \)
53 \( 1 + (3.75 + 3.75i)T + 53iT^{2} \)
59 \( 1 + (3.74 + 3.74i)T + 59iT^{2} \)
61 \( 1 + (5.14 - 5.14i)T - 61iT^{2} \)
67 \( 1 + (-5.38 + 5.38i)T - 67iT^{2} \)
71 \( 1 + 5.68iT - 71T^{2} \)
73 \( 1 - 6.59iT - 73T^{2} \)
79 \( 1 - 2.47T + 79T^{2} \)
83 \( 1 + (-1.59 + 1.59i)T - 83iT^{2} \)
89 \( 1 + 0.601iT - 89T^{2} \)
97 \( 1 - 12.6T + 97T^{2} \)
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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

−10.76827051855155813024450573404, −9.959637232902871002764235842377, −9.245767063021448833329655142775, −8.106254471488238150392097127943, −7.68861573178103780379130363289, −6.60237527793501363625390272541, −5.67742674069602776183356326096, −4.79295638928941717239511554678, −2.83947493829870582470501010308, −1.65526303282506673156491146832, 0.74326123150426441123616521677, 2.40921355352352275455544933443, 3.51485977148030481762436470944, 4.58780621628511777166060912432, 5.96162685936099728694999374140, 7.29592974959377413984659149044, 7.953477209065323754280626130570, 9.016854286475700014345454308609, 9.752168591663261804487439984214, 10.23468882086503646684381707287

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