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Properties

Label 2-2240-140.139-c1-0-57
Degree $2$
Conductor $2240$
Sign $0.987 + 0.155i$
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  + 0.662i·3-s + (1.31 − 1.81i)5-s + (1.19 + 2.35i)7-s + 2.56·9-s − 3.09i·11-s − 4.66·13-s + (1.19 + 0.868i)15-s − 2.04·17-s + 5.60·19-s + (−1.56 + 0.794i)21-s + 1.87·23-s + (−1.56 − 4.74i)25-s + 3.68i·27-s + 3.56·29-s + 8.74·31-s + ⋯
L(s)  = 1  + 0.382i·3-s + (0.586 − 0.810i)5-s + (0.453 + 0.891i)7-s + 0.853·9-s − 0.932i·11-s − 1.29·13-s + (0.309 + 0.224i)15-s − 0.496·17-s + 1.28·19-s + (−0.340 + 0.173i)21-s + 0.390·23-s + (−0.312 − 0.949i)25-s + 0.708i·27-s + 0.661·29-s + 1.57·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.155i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 + 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $0.987 + 0.155i$
Analytic conductor: \(17.8864\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (2239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :1/2),\ 0.987 + 0.155i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.246344880\)
\(L(\frac12)\) \(\approx\) \(2.246344880\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.31 + 1.81i)T \)
7 \( 1 + (-1.19 - 2.35i)T \)
good3 \( 1 - 0.662iT - 3T^{2} \)
11 \( 1 + 3.09iT - 11T^{2} \)
13 \( 1 + 4.66T + 13T^{2} \)
17 \( 1 + 2.04T + 17T^{2} \)
19 \( 1 - 5.60T + 19T^{2} \)
23 \( 1 - 1.87T + 23T^{2} \)
29 \( 1 - 3.56T + 29T^{2} \)
31 \( 1 - 8.74T + 31T^{2} \)
37 \( 1 - 3.70iT - 37T^{2} \)
41 \( 1 + 8.48iT - 41T^{2} \)
43 \( 1 - 4.27T + 43T^{2} \)
47 \( 1 - 0.290iT - 47T^{2} \)
53 \( 1 + 9.49iT - 53T^{2} \)
59 \( 1 + 8.05T + 59T^{2} \)
61 \( 1 - 6.45iT - 61T^{2} \)
67 \( 1 - 2.39T + 67T^{2} \)
71 \( 1 + 9.65iT - 71T^{2} \)
73 \( 1 + 4.09T + 73T^{2} \)
79 \( 1 + 1.35iT - 79T^{2} \)
83 \( 1 - 12.4iT - 83T^{2} \)
89 \( 1 - 2.82iT - 89T^{2} \)
97 \( 1 - 6.14T + 97T^{2} \)
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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

−9.076614959575951966400224547429, −8.412224169640274440080620100197, −7.59156936859556467613879330334, −6.59876646135787959391857100686, −5.65607835626401501788952040684, −4.98629042497208890094942774405, −4.48014034077230988533061019803, −3.10004760685678598730541864823, −2.14352590811179533898667439843, −0.950947964654858878993939501751, 1.13333879925596055205942680181, 2.15694906467915735955274502604, 3.06601580530008338380662262079, 4.44415074027392282024389555905, 4.83556241963790239517210411027, 6.12312503530759400022357062681, 7.01703254219672185516409801348, 7.30126249731937387140967798023, 7.934053359424858755307487463157, 9.326065861431230739870811486253

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