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Properties

Label 2-560-20.7-c1-0-0
Degree $2$
Conductor $560$
Sign $-0.619 - 0.784i$
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  + (−0.964 − 0.964i)3-s + (−1.75 − 1.38i)5-s + (−0.707 + 0.707i)7-s − 1.14i·9-s + 3.37i·11-s + (0.610 − 0.610i)13-s + (0.350 + 3.02i)15-s + (−2.11 − 2.11i)17-s − 7.98·19-s + 1.36·21-s + (3.02 + 3.02i)23-s + (1.14 + 4.86i)25-s + (−3.99 + 3.99i)27-s + 9.15i·29-s + 8.68i·31-s + ⋯
L(s)  = 1  + (−0.556 − 0.556i)3-s + (−0.783 − 0.621i)5-s + (−0.267 + 0.267i)7-s − 0.380i·9-s + 1.01i·11-s + (0.169 − 0.169i)13-s + (0.0904 + 0.781i)15-s + (−0.513 − 0.513i)17-s − 1.83·19-s + 0.297·21-s + (0.631 + 0.631i)23-s + (0.228 + 0.973i)25-s + (−0.768 + 0.768i)27-s + 1.69i·29-s + 1.55i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.619 - 0.784i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.619 - 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $-0.619 - 0.784i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ -0.619 - 0.784i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0405601 + 0.0837441i\)
\(L(\frac12)\) \(\approx\) \(0.0405601 + 0.0837441i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.75 + 1.38i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
good3 \( 1 + (0.964 + 0.964i)T + 3iT^{2} \)
11 \( 1 - 3.37iT - 11T^{2} \)
13 \( 1 + (-0.610 + 0.610i)T - 13iT^{2} \)
17 \( 1 + (2.11 + 2.11i)T + 17iT^{2} \)
19 \( 1 + 7.98T + 19T^{2} \)
23 \( 1 + (-3.02 - 3.02i)T + 23iT^{2} \)
29 \( 1 - 9.15iT - 29T^{2} \)
31 \( 1 - 8.68iT - 31T^{2} \)
37 \( 1 + (4.36 + 4.36i)T + 37iT^{2} \)
41 \( 1 - 2.23T + 41T^{2} \)
43 \( 1 + (8.33 + 8.33i)T + 43iT^{2} \)
47 \( 1 + (-1.66 + 1.66i)T - 47iT^{2} \)
53 \( 1 + (-0.221 + 0.221i)T - 53iT^{2} \)
59 \( 1 - 1.22T + 59T^{2} \)
61 \( 1 + 5.55T + 61T^{2} \)
67 \( 1 + (-3.50 + 3.50i)T - 67iT^{2} \)
71 \( 1 - 3.85iT - 71T^{2} \)
73 \( 1 + (3.86 - 3.86i)T - 73iT^{2} \)
79 \( 1 + 7.93T + 79T^{2} \)
83 \( 1 + (5.65 + 5.65i)T + 83iT^{2} \)
89 \( 1 + 15.0iT - 89T^{2} \)
97 \( 1 + (9.17 + 9.17i)T + 97iT^{2} \)
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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

−11.22001820256045777245743457558, −10.35108387955328796943352243846, −9.028486093449200664527333792358, −8.627192437349641438149435921708, −7.14322275197848546838293040169, −6.85978621284986081395265335827, −5.50266341828017490009869363684, −4.59588659333248936389741382089, −3.41141231038175925360581375390, −1.64400802600488165058544735022, 0.05593820566565987618826834425, 2.53300275866004390180837157856, 3.92413494331382483126461476960, 4.53629849046631569384984630503, 6.03093118497755643140699418003, 6.58372242161567002273687391818, 7.931160890160987189432634475215, 8.507053173168121575241616212810, 9.832932036659348979048608747611, 10.75357004406765707727857451832

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