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Properties

Label 2-2240-16.13-c1-0-24
Degree $2$
Conductor $2240$
Sign $0.775 + 0.631i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.989 − 0.989i)3-s + (0.707 − 0.707i)5-s + i·7-s − 1.04i·9-s + (3.71 − 3.71i)11-s + (2.81 + 2.81i)13-s − 1.39·15-s + 0.895·17-s + (3.33 + 3.33i)19-s + (0.989 − 0.989i)21-s + 6.38i·23-s − 1.00i·25-s + (−3.99 + 3.99i)27-s + (1.18 + 1.18i)29-s + 6.25·31-s + ⋯
L(s)  = 1  + (−0.571 − 0.571i)3-s + (0.316 − 0.316i)5-s + 0.377i·7-s − 0.346i·9-s + (1.12 − 1.12i)11-s + (0.780 + 0.780i)13-s − 0.361·15-s + 0.217·17-s + (0.765 + 0.765i)19-s + (0.215 − 0.215i)21-s + 1.33i·23-s − 0.200i·25-s + (−0.769 + 0.769i)27-s + (0.219 + 0.219i)29-s + 1.12·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 + 0.631i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.775 + 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $0.775 + 0.631i$
Analytic conductor: \(17.8864\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (1681, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :1/2),\ 0.775 + 0.631i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.794792037\)
\(L(\frac12)\) \(\approx\) \(1.794792037\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.989 + 0.989i)T + 3iT^{2} \)
11 \( 1 + (-3.71 + 3.71i)T - 11iT^{2} \)
13 \( 1 + (-2.81 - 2.81i)T + 13iT^{2} \)
17 \( 1 - 0.895T + 17T^{2} \)
19 \( 1 + (-3.33 - 3.33i)T + 19iT^{2} \)
23 \( 1 - 6.38iT - 23T^{2} \)
29 \( 1 + (-1.18 - 1.18i)T + 29iT^{2} \)
31 \( 1 - 6.25T + 31T^{2} \)
37 \( 1 + (4.00 - 4.00i)T - 37iT^{2} \)
41 \( 1 + 2.72iT - 41T^{2} \)
43 \( 1 + (-8.61 + 8.61i)T - 43iT^{2} \)
47 \( 1 + 3.21T + 47T^{2} \)
53 \( 1 + (3.27 - 3.27i)T - 53iT^{2} \)
59 \( 1 + (7.51 - 7.51i)T - 59iT^{2} \)
61 \( 1 + (-0.859 - 0.859i)T + 61iT^{2} \)
67 \( 1 + (-7.29 - 7.29i)T + 67iT^{2} \)
71 \( 1 + 5.87iT - 71T^{2} \)
73 \( 1 + 14.4iT - 73T^{2} \)
79 \( 1 - 4.29T + 79T^{2} \)
83 \( 1 + (-10.3 - 10.3i)T + 83iT^{2} \)
89 \( 1 - 0.769iT - 89T^{2} \)
97 \( 1 - 16.7T + 97T^{2} \)
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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

−9.090230238466406224242455335649, −8.256271132056861253390344486458, −7.27525473281519846777199347098, −6.35413340997950329520887085104, −6.01388069478784334383695117090, −5.25252038950820742365787311810, −3.93232568751099801530797756927, −3.27412786267934133206735245333, −1.63644485970848651950111190050, −0.978192660353779711581100735246, 0.965518660887989484589420509399, 2.30644565495871769268009071889, 3.45916459446844943198684229902, 4.47245528046319710866353529493, 4.96542579260559230352381834506, 6.07687059179365906134872889658, 6.61962047237756660150001122124, 7.53808526794800055573198023381, 8.323699883970295612275678203535, 9.400073409085242919540293933755

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