Properties

Label 2-2240-28.27-c1-0-25
Degree $2$
Conductor $2240$
Sign $0.954 - 0.299i$
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  − 0.792·3-s i·5-s + (0.792 + 2.52i)7-s − 2.37·9-s + 0.792i·11-s − 5.37i·13-s + 0.792i·15-s + 3.37i·17-s − 3.46·19-s + (−0.627 − 2i)21-s − 1.87i·23-s − 25-s + 4.25·27-s + 5.37·29-s + 8.51·31-s + ⋯
L(s)  = 1  − 0.457·3-s − 0.447i·5-s + (0.299 + 0.954i)7-s − 0.790·9-s + 0.238i·11-s − 1.49i·13-s + 0.204i·15-s + 0.817i·17-s − 0.794·19-s + (−0.136 − 0.436i)21-s − 0.391i·23-s − 0.200·25-s + 0.819·27-s + 0.997·29-s + 1.52·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 - 0.299i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.954 - 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $0.954 - 0.299i$
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: \((2,\ 2240,\ (\ :1/2),\ 0.954 - 0.299i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.303514869\)
\(L(\frac12)\) \(\approx\) \(1.303514869\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.792 - 2.52i)T \)
good3 \( 1 + 0.792T + 3T^{2} \)
11 \( 1 - 0.792iT - 11T^{2} \)
13 \( 1 + 5.37iT - 13T^{2} \)
17 \( 1 - 3.37iT - 17T^{2} \)
19 \( 1 + 3.46T + 19T^{2} \)
23 \( 1 + 1.87iT - 23T^{2} \)
29 \( 1 - 5.37T + 29T^{2} \)
31 \( 1 - 8.51T + 31T^{2} \)
37 \( 1 - 0.744T + 37T^{2} \)
41 \( 1 - 2.74iT - 41T^{2} \)
43 \( 1 - 3.46iT - 43T^{2} \)
47 \( 1 - 11.1T + 47T^{2} \)
53 \( 1 + 11.4T + 53T^{2} \)
59 \( 1 - 6.63T + 59T^{2} \)
61 \( 1 + 0.744iT - 61T^{2} \)
67 \( 1 - 6.63iT - 67T^{2} \)
71 \( 1 + 6.63iT - 71T^{2} \)
73 \( 1 - 2.74iT - 73T^{2} \)
79 \( 1 - 14.0iT - 79T^{2} \)
83 \( 1 - 10.3T + 83T^{2} \)
89 \( 1 + 17.4iT - 89T^{2} \)
97 \( 1 + 2.11iT - 97T^{2} \)
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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

−8.797212145508855441642573701777, −8.390998312995158025781756971942, −7.80944796377549890346320843846, −6.37008235213713745979460702950, −5.97894915154398689038260947367, −5.15003513863547773879473108767, −4.48177369057433092860007402443, −3.11030484287245875526584759119, −2.31263021664597465325711340608, −0.831847029922086119251576576787, 0.68152847842491732306821513207, 2.12617256126189113206418810041, 3.19808507581188066338169505267, 4.27872688503216016413482680368, 4.87484315464577726723964952278, 6.04041983739977062255650789683, 6.64387636834882795977184001054, 7.29670550033201254443580186238, 8.244691146087565723537877100858, 8.956911605221338820573778391631

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