Properties

Label 2-2240-40.29-c1-0-56
Degree $2$
Conductor $2240$
Sign $0.0650 + 0.997i$
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  + 1.29·3-s + (0.437 − 2.19i)5-s + i·7-s − 1.32·9-s + 0.711i·11-s + 1.85·13-s + (0.564 − 2.83i)15-s + 3.75i·17-s − 7.57i·19-s + 1.29i·21-s − 8.45i·23-s + (−4.61 − 1.91i)25-s − 5.59·27-s − 5.24i·29-s + 0.349·31-s + ⋯
L(s)  = 1  + 0.746·3-s + (0.195 − 0.980i)5-s + 0.377i·7-s − 0.443·9-s + 0.214i·11-s + 0.514·13-s + (0.145 − 0.731i)15-s + 0.911i·17-s − 1.73i·19-s + 0.282i·21-s − 1.76i·23-s + (−0.923 − 0.383i)25-s − 1.07·27-s − 0.974i·29-s + 0.0626·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0650 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0650 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $0.0650 + 0.997i$
Analytic conductor: \(17.8864\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (1569, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :1/2),\ 0.0650 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.017337926\)
\(L(\frac12)\) \(\approx\) \(2.017337926\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.437 + 2.19i)T \)
7 \( 1 - iT \)
good3 \( 1 - 1.29T + 3T^{2} \)
11 \( 1 - 0.711iT - 11T^{2} \)
13 \( 1 - 1.85T + 13T^{2} \)
17 \( 1 - 3.75iT - 17T^{2} \)
19 \( 1 + 7.57iT - 19T^{2} \)
23 \( 1 + 8.45iT - 23T^{2} \)
29 \( 1 + 5.24iT - 29T^{2} \)
31 \( 1 - 0.349T + 31T^{2} \)
37 \( 1 - 7.50T + 37T^{2} \)
41 \( 1 - 2.74T + 41T^{2} \)
43 \( 1 - 2.07T + 43T^{2} \)
47 \( 1 + 4.69iT - 47T^{2} \)
53 \( 1 - 9.81T + 53T^{2} \)
59 \( 1 - 6.54iT - 59T^{2} \)
61 \( 1 + 6.50iT - 61T^{2} \)
67 \( 1 + 10.4T + 67T^{2} \)
71 \( 1 - 5.53T + 71T^{2} \)
73 \( 1 + 9.10iT - 73T^{2} \)
79 \( 1 + 4.54T + 79T^{2} \)
83 \( 1 + 4.92T + 83T^{2} \)
89 \( 1 + 5.58T + 89T^{2} \)
97 \( 1 + 14.1iT - 97T^{2} \)
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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

−8.717958196838985241241688887468, −8.422076214462040536414400845380, −7.53022658783037603335210042124, −6.36466252096229781803478922006, −5.75972198328856653101920358442, −4.70273409192311851215615724018, −4.06654828077644163623313132746, −2.78287937779449097291159160421, −2.10736749563108261626133635289, −0.62869529879668265166693025718, 1.45627239458198185133106979147, 2.64366503574789659870415508140, 3.37519821660085464249300340643, 4.01293789254082938299592383056, 5.52840086751224400866706167530, 5.99968448865864889042198351789, 7.10672475325038126676249993489, 7.67212414836594734942344584083, 8.351254777809505601713452710880, 9.331014038297916234741856383214

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