↑

Properties

Label 2-560-16.5-c1-0-18
Degree $2$
Conductor $560$
Sign $-0.0421 - 0.999i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.40 + 0.123i)2-s + (−1.35 + 1.35i)3-s + (1.96 + 0.348i)4-s + (−0.707 − 0.707i)5-s + (−2.08 + 1.74i)6-s + i·7-s + (2.73 + 0.734i)8-s − 0.692i·9-s + (−0.908 − 1.08i)10-s + (2.35 + 2.35i)11-s + (−3.14 + 2.20i)12-s + (−0.605 + 0.605i)13-s + (−0.123 + 1.40i)14-s + 1.92·15-s + (3.75 + 1.37i)16-s − 1.39·17-s + ⋯
L(s)  = 1  + (0.996 + 0.0875i)2-s + (−0.784 + 0.784i)3-s + (0.984 + 0.174i)4-s + (−0.316 − 0.316i)5-s + (−0.850 + 0.712i)6-s + 0.377i·7-s + (0.965 + 0.259i)8-s − 0.230i·9-s + (−0.287 − 0.342i)10-s + (0.709 + 0.709i)11-s + (−0.909 + 0.635i)12-s + (−0.168 + 0.168i)13-s + (−0.0330 + 0.376i)14-s + 0.496·15-s + (0.939 + 0.343i)16-s − 0.339·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0421 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0421 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $-0.0421 - 0.999i$
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: \((2,\ 560,\ (\ :1/2),\ -0.0421 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36647 + 1.42539i\)
\(L(\frac12)\) \(\approx\) \(1.36647 + 1.42539i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 - 0.123i)T \)
5 \( 1 + (0.707 + 0.707i)T \)
7 \( 1 - iT \)
good3 \( 1 + (1.35 - 1.35i)T - 3iT^{2} \)
11 \( 1 + (-2.35 - 2.35i)T + 11iT^{2} \)
13 \( 1 + (0.605 - 0.605i)T - 13iT^{2} \)
17 \( 1 + 1.39T + 17T^{2} \)
19 \( 1 + (2.68 - 2.68i)T - 19iT^{2} \)
23 \( 1 - 5.65iT - 23T^{2} \)
29 \( 1 + (0.938 - 0.938i)T - 29iT^{2} \)
31 \( 1 - 8.67T + 31T^{2} \)
37 \( 1 + (1.49 + 1.49i)T + 37iT^{2} \)
41 \( 1 + 6.63iT - 41T^{2} \)
43 \( 1 + (2.62 + 2.62i)T + 43iT^{2} \)
47 \( 1 - 7.68T + 47T^{2} \)
53 \( 1 + (6.01 + 6.01i)T + 53iT^{2} \)
59 \( 1 + (-4.55 - 4.55i)T + 59iT^{2} \)
61 \( 1 + (-1.47 + 1.47i)T - 61iT^{2} \)
67 \( 1 + (-4.96 + 4.96i)T - 67iT^{2} \)
71 \( 1 + 11.4iT - 71T^{2} \)
73 \( 1 + 0.758iT - 73T^{2} \)
79 \( 1 - 14.3T + 79T^{2} \)
83 \( 1 + (8.55 - 8.55i)T - 83iT^{2} \)
89 \( 1 + 7.52iT - 89T^{2} \)
97 \( 1 - 0.0599T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.18484550293684047923981810740, −10.35139948649415638084015321465, −9.458275275650571855012193349867, −8.217842860884646227640518402730, −7.12497003584724304393802379364, −6.13991960779802683825374119475, −5.25433664873546264912246157824, −4.48293484019998777305395456974, −3.69531035533978357105438748460, −1.99682881477320552838997630145, 0.950440951666691266471053618976, 2.63051845693396477937012123656, 3.90640873100414373526498199436, 4.89519786267706846831224343866, 6.29909930215090066429143793122, 6.46437697515880599254961414308, 7.44637957817809524464910201012, 8.549948962272458921784148124673, 10.04398080501267172824178018368, 11.02293360229074971227088780751

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