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Properties

Label 2-2240-28.27-c1-0-47
Degree $2$
Conductor $2240$
Sign $0.755 + 0.654i$
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  + 1.73·3-s + i·5-s + (−1.73 + 2i)7-s − 3.73i·11-s − 6.46i·13-s + 1.73i·15-s + 0.464i·17-s + 6·19-s + (−2.99 + 3.46i)21-s − 5.46i·23-s − 25-s − 5.19·27-s + 5.92·29-s + 6·31-s − 6.46i·33-s + ⋯
L(s)  = 1  + 1.00·3-s + 0.447i·5-s + (−0.654 + 0.755i)7-s − 1.12i·11-s − 1.79i·13-s + 0.447i·15-s + 0.112i·17-s + 1.37·19-s + (−0.654 + 0.755i)21-s − 1.13i·23-s − 0.200·25-s − 1.00·27-s + 1.10·29-s + 1.07·31-s − 1.12i·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $0.755 + 0.654i$
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: \((2,\ 2240,\ (\ :1/2),\ 0.755 + 0.654i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.174128165\)
\(L(\frac12)\) \(\approx\) \(2.174128165\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.73 - 2i)T \)
good3 \( 1 - 1.73T + 3T^{2} \)
11 \( 1 + 3.73iT - 11T^{2} \)
13 \( 1 + 6.46iT - 13T^{2} \)
17 \( 1 - 0.464iT - 17T^{2} \)
19 \( 1 - 6T + 19T^{2} \)
23 \( 1 + 5.46iT - 23T^{2} \)
29 \( 1 - 5.92T + 29T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 - 2.53T + 37T^{2} \)
41 \( 1 - 3.46iT - 41T^{2} \)
43 \( 1 + 2iT - 43T^{2} \)
47 \( 1 - 1.73T + 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 - 3.46T + 59T^{2} \)
61 \( 1 + 2.53iT - 61T^{2} \)
67 \( 1 + 3.46iT - 67T^{2} \)
71 \( 1 - 0.535iT - 71T^{2} \)
73 \( 1 + 0.928iT - 73T^{2} \)
79 \( 1 + 2.66iT - 79T^{2} \)
83 \( 1 - 8.53T + 83T^{2} \)
89 \( 1 - 9.46iT - 89T^{2} \)
97 \( 1 - 7.39iT - 97T^{2} \)
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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

−8.778553202904906031659095410025, −8.238458996622628089679563516999, −7.72543747416526908182652832786, −6.50364699815216869252605295754, −5.88621325116723793911903747206, −5.09980817992872615896548931574, −3.59379938497821216555033148530, −2.93844212012943473120486048310, −2.63200233559675387129061570578, −0.71231489609207408159789625810, 1.26947934288390833145307867582, 2.35225746817288645667600633964, 3.36611593148128421952292254439, 4.17841848431656467640027871152, 4.89582677063930412665219407986, 6.12767531458918756326959362014, 7.07266397552435408721005352964, 7.49620955589930771958085253012, 8.419811402211395606897172284509, 9.352476670649005870590538492705

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