Properties

Label 2-2240-5.4-c1-0-60
Degree $2$
Conductor $2240$
Sign $0.447 + 0.894i$
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  + i·3-s + (2 − i)5-s i·7-s + 2·9-s − 3·11-s i·13-s + (1 + 2i)15-s − 7i·17-s + 21-s − 6i·23-s + (3 − 4i)25-s + 5i·27-s − 5·29-s − 2·31-s − 3i·33-s + ⋯
L(s)  = 1  + 0.577i·3-s + (0.894 − 0.447i)5-s − 0.377i·7-s + 0.666·9-s − 0.904·11-s − 0.277i·13-s + (0.258 + 0.516i)15-s − 1.69i·17-s + 0.218·21-s − 1.25i·23-s + (0.600 − 0.800i)25-s + 0.962i·27-s − 0.928·29-s − 0.359·31-s − 0.522i·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(17.8864\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :1/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.855816271\)
\(L(\frac12)\) \(\approx\) \(1.855816271\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 + i)T \)
7 \( 1 + iT \)
good3 \( 1 - iT - 3T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 + iT - 13T^{2} \)
17 \( 1 + 7iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 + 5T + 29T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 + 3iT - 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + 10T + 59T^{2} \)
61 \( 1 - 8T + 61T^{2} \)
67 \( 1 + 2iT - 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 - 6iT - 73T^{2} \)
79 \( 1 + 5T + 79T^{2} \)
83 \( 1 + 4iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 7iT - 97T^{2} \)
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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

−9.145210060327404657014522046302, −8.192152263829947548463624324076, −7.31021401351418615930984807572, −6.61934665433711149719088703750, −5.43752675546078448855074798468, −5.00594796657550242777241291339, −4.19175140845424130484500144937, −3.00692436585414587288626000558, −2.03873493160167086590209888837, −0.62456860391467308714401548794, 1.51795013975538364452476919141, 2.11849522457810324846160212487, 3.25072636649214047144377526473, 4.32814274497728760755599752704, 5.54291837291380504301698890474, 5.96422198649062102043075658992, 6.86159234660814441517529223325, 7.57654844002445305010518552799, 8.265151592797047247579718580752, 9.324436535970647935784959557627

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