Properties

Label 2-2240-28.27-c1-0-35
Degree $2$
Conductor $2240$
Sign $0.761 - 0.648i$
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  + 3.29·3-s i·5-s + (−2.01 + 1.71i)7-s + 7.85·9-s + 1.43i·11-s + 5.60i·13-s − 3.29i·15-s + 1.85i·17-s − 3.04·19-s + (−6.64 + 5.65i)21-s − 0.989i·23-s − 25-s + 16.0·27-s + 5.89·29-s + 7.82·31-s + ⋯
L(s)  = 1  + 1.90·3-s − 0.447i·5-s + (−0.761 + 0.648i)7-s + 2.61·9-s + 0.432i·11-s + 1.55i·13-s − 0.850i·15-s + 0.450i·17-s − 0.697·19-s + (−1.44 + 1.23i)21-s − 0.206i·23-s − 0.200·25-s + 3.08·27-s + 1.09·29-s + 1.40·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.761 - 0.648i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.761 - 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $0.761 - 0.648i$
Analytic conductor: \(17.8864\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (1791, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :1/2),\ 0.761 - 0.648i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.366954492\)
\(L(\frac12)\) \(\approx\) \(3.366954492\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
7 \( 1 + (2.01 - 1.71i)T \)
good3 \( 1 - 3.29T + 3T^{2} \)
11 \( 1 - 1.43iT - 11T^{2} \)
13 \( 1 - 5.60iT - 13T^{2} \)
17 \( 1 - 1.85iT - 17T^{2} \)
19 \( 1 + 3.04T + 19T^{2} \)
23 \( 1 + 0.989iT - 23T^{2} \)
29 \( 1 - 5.89T + 29T^{2} \)
31 \( 1 - 7.82T + 31T^{2} \)
37 \( 1 - 3.93T + 37T^{2} \)
41 \( 1 - 8.37iT - 41T^{2} \)
43 \( 1 - 0.890iT - 43T^{2} \)
47 \( 1 - 0.347T + 47T^{2} \)
53 \( 1 + 12.1T + 53T^{2} \)
59 \( 1 - 4.32T + 59T^{2} \)
61 \( 1 + 13.2iT - 61T^{2} \)
67 \( 1 + 15.3iT - 67T^{2} \)
71 \( 1 + 0.574iT - 71T^{2} \)
73 \( 1 - 13.6iT - 73T^{2} \)
79 \( 1 + 3.17iT - 79T^{2} \)
83 \( 1 + 7.59T + 83T^{2} \)
89 \( 1 - 8.08iT - 89T^{2} \)
97 \( 1 + 4.66iT - 97T^{2} \)
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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

−9.107936123606531112979115012799, −8.378263328850732258695191877085, −7.967880024487965223952187387542, −6.72577122843171287211499205573, −6.41401868232627919357793997438, −4.68570514398953192510245798599, −4.22505458571405684978759224838, −3.17385549796772652908787692448, −2.39343793582761918065164248261, −1.57389112190683697171588991665, 0.954667450777090617190653112092, 2.54773766449097333669383398078, 2.99981153221213990465384269243, 3.73415146967024992819473965198, 4.59804166093889779166644606759, 6.00977614177082383667586436902, 6.90499128364019151679483380841, 7.55973259112728776739863898843, 8.221810673196098827362096234241, 8.802678278195641759560329615697

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