↑

Properties

Label 2-2240-16.5-c1-0-3
Degree $2$
Conductor $2240$
Sign $-0.752 - 0.658i$
Analytic cond. $17.8864$
Root an. cond. $4.22924$
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  + (−2.18 + 2.18i)3-s + (−0.707 − 0.707i)5-s − i·7-s − 6.57i·9-s + (−0.847 − 0.847i)11-s + (3.07 − 3.07i)13-s + 3.09·15-s − 5.75·17-s + (−3.49 + 3.49i)19-s + (2.18 + 2.18i)21-s − 8.84i·23-s + 1.00i·25-s + (7.82 + 7.82i)27-s + (−2.20 + 2.20i)29-s + 5.44·31-s + ⋯
L(s)  = 1  + (−1.26 + 1.26i)3-s + (−0.316 − 0.316i)5-s − 0.377i·7-s − 2.19i·9-s + (−0.255 − 0.255i)11-s + (0.853 − 0.853i)13-s + 0.799·15-s − 1.39·17-s + (−0.802 + 0.802i)19-s + (0.477 + 0.477i)21-s − 1.84i·23-s + 0.200i·25-s + (1.50 + 1.50i)27-s + (−0.409 + 0.409i)29-s + 0.978·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.752 - 0.658i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.752 - 0.658i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $-0.752 - 0.658i$
Analytic conductor: \(17.8864\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (561, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :1/2),\ -0.752 - 0.658i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4167805511\)
\(L(\frac12)\) \(\approx\) \(0.4167805511\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 + 0.707i)T \)
7 \( 1 + iT \)
good3 \( 1 + (2.18 - 2.18i)T - 3iT^{2} \)
11 \( 1 + (0.847 + 0.847i)T + 11iT^{2} \)
13 \( 1 + (-3.07 + 3.07i)T - 13iT^{2} \)
17 \( 1 + 5.75T + 17T^{2} \)
19 \( 1 + (3.49 - 3.49i)T - 19iT^{2} \)
23 \( 1 + 8.84iT - 23T^{2} \)
29 \( 1 + (2.20 - 2.20i)T - 29iT^{2} \)
31 \( 1 - 5.44T + 31T^{2} \)
37 \( 1 + (-4.16 - 4.16i)T + 37iT^{2} \)
41 \( 1 - 4.94iT - 41T^{2} \)
43 \( 1 + (-7.34 - 7.34i)T + 43iT^{2} \)
47 \( 1 + 5.42T + 47T^{2} \)
53 \( 1 + (2.14 + 2.14i)T + 53iT^{2} \)
59 \( 1 + (-3.49 - 3.49i)T + 59iT^{2} \)
61 \( 1 + (3.45 - 3.45i)T - 61iT^{2} \)
67 \( 1 + (3.12 - 3.12i)T - 67iT^{2} \)
71 \( 1 + 0.245iT - 71T^{2} \)
73 \( 1 - 0.160iT - 73T^{2} \)
79 \( 1 - 9.95T + 79T^{2} \)
83 \( 1 + (6.68 - 6.68i)T - 83iT^{2} \)
89 \( 1 - 6.01iT - 89T^{2} \)
97 \( 1 + 14.8T + 97T^{2} \)
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

−9.491418217294295973970449591936, −8.583054126818158603421242006827, −8.034458268315875305002808033004, −6.50245655408092090248911609754, −6.26421571293491068187189852973, −5.25883057482757319131191417598, −4.41422417193252851561425699646, −4.08461162172323585447195937652, −2.87195762294301449404716353539, −0.921526709435038689111909602102, 0.22045814985880021701548956573, 1.66897695852791772540382023360, 2.42845050776319529981328575977, 3.99752073687393081083169730461, 4.89981358190381498522706766283, 5.85668305716609199671328933162, 6.41337792205759070549502881430, 7.06012785326649450799146851813, 7.68236630324951557198911208405, 8.628345414262880615372897549551

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