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Properties

Label 2-2240-140.139-c1-0-1
Degree $2$
Conductor $2240$
Sign $-i$
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  − 3.25i·3-s + 2.23·5-s + 2.64i·7-s − 7.62·9-s + 0.359i·11-s − 5.64·13-s − 7.28i·15-s − 7.99·17-s + 8.62·21-s + 5.00·25-s + 15.0i·27-s + 0.623·29-s + 1.17·33-s + 5.91i·35-s + 18.4i·39-s + ⋯
L(s)  = 1  − 1.88i·3-s + 0.999·5-s + 0.999i·7-s − 2.54·9-s + 0.108i·11-s − 1.56·13-s − 1.88i·15-s − 1.93·17-s + 1.88·21-s + 1.00·25-s + 2.90i·27-s + 0.115·29-s + 0.204·33-s + 0.999i·35-s + 2.94i·39-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(0.1932821272\)
\(L(\frac12)\) \(\approx\) \(0.1932821272\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.23T \)
7 \( 1 - 2.64iT \)
good3 \( 1 + 3.25iT - 3T^{2} \)
11 \( 1 - 0.359iT - 11T^{2} \)
13 \( 1 + 5.64T + 13T^{2} \)
17 \( 1 + 7.99T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 0.623T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 5.71iT - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 - 11.8iT - 71T^{2} \)
73 \( 1 + 13.4T + 73T^{2} \)
79 \( 1 - 8.00iT - 79T^{2} \)
83 \( 1 + 15.8iT - 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + 12.6T + 97T^{2} \)
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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

−9.002211632337251757707851655960, −8.491993168211965615466158240395, −7.50990297693706581553477547216, −6.83690737251356038265188932178, −6.28227463880016712761291400072, −5.54575336995425253081820536709, −4.74052212260564824426495591270, −2.66415773827890002852940406719, −2.42441560630674281224211086841, −1.55827402865389829302806693561, 0.05859793494028838255593766708, 2.20360587676510659218615833388, 3.09061268317235211881963558598, 4.23097769096637896235806967305, 4.68896149838753873142651403825, 5.38305831793971773886313657366, 6.38287134751296464442386314388, 7.19402322308151678642933764507, 8.420548021844602590941216379255, 9.136093902613381764746828025447

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