Properties

Label 2-560-140.139-c1-0-12
Degree $2$
Conductor $560$
Sign $0.450 - 0.892i$
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  + 1.41i·3-s + (1.22 + 1.87i)5-s + 2.64·7-s + 0.999·9-s − 3.46i·11-s + 2.44·13-s + (−2.64 + 1.73i)15-s + 4.89·17-s − 6.48·19-s + 3.74i·21-s + (−2 + 4.58i)25-s + 5.65i·27-s − 6·29-s + 4.89·33-s + (3.24 + 4.94i)35-s + ⋯
L(s)  = 1  + 0.816i·3-s + (0.547 + 0.836i)5-s + 0.999·7-s + 0.333·9-s − 1.04i·11-s + 0.679·13-s + (−0.683 + 0.447i)15-s + 1.18·17-s − 1.48·19-s + 0.816i·21-s + (−0.400 + 0.916i)25-s + 1.08i·27-s − 1.11·29-s + 0.852·33-s + (0.547 + 0.836i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.450 - 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.450 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.450 - 0.892i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ 0.450 - 0.892i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.57795 + 0.970971i\)
\(L(\frac12)\) \(\approx\) \(1.57795 + 0.970971i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.22 - 1.87i)T \)
7 \( 1 - 2.64T \)
good3 \( 1 - 1.41iT - 3T^{2} \)
11 \( 1 + 3.46iT - 11T^{2} \)
13 \( 1 - 2.44T + 13T^{2} \)
17 \( 1 - 4.89T + 17T^{2} \)
19 \( 1 + 6.48T + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 9.16iT - 37T^{2} \)
41 \( 1 - 7.48iT - 41T^{2} \)
43 \( 1 + 5.29T + 43T^{2} \)
47 \( 1 + 2.82iT - 47T^{2} \)
53 \( 1 - 9.16iT - 53T^{2} \)
59 \( 1 + 6.48T + 59T^{2} \)
61 \( 1 + 11.2iT - 61T^{2} \)
67 \( 1 - 5.29T + 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + 9.79T + 73T^{2} \)
79 \( 1 + 6.92iT - 79T^{2} \)
83 \( 1 - 9.89iT - 83T^{2} \)
89 \( 1 + 7.48iT - 89T^{2} \)
97 \( 1 + 14.6T + 97T^{2} \)
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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

−10.91532993145059968693917743221, −10.21129077007459891533692131732, −9.266558772274155164827877681063, −8.347982656869909109069162582559, −7.39399605529486446347427078598, −6.17234452191215073281985829542, −5.41866201893728592316970859352, −4.18494849367790101095004057903, −3.26216081071959783349691340205, −1.71364936448650425198429667277, 1.34047748968225242711654782314, 2.05196576390523542014259789434, 4.10661955179690778677668997076, 5.01523587055777003121707448605, 6.03763820338653779372964590731, 7.08572601943665368229634578987, 7.989682009198410559985515891577, 8.642006082051091656129750836681, 9.766426521974510353203333181079, 10.52080392241116934419607118667

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