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Properties

Label 2-2240-280.139-c1-0-61
Degree $2$
Conductor $2240$
Sign $0.707 - 0.707i$
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  + 2.23·3-s + 2.23i·5-s + 2.64i·7-s + 2.00·9-s + 5.91·11-s − 6.70i·13-s + 5.00i·15-s + 7.93·17-s + 5.91i·21-s − 5.00·25-s − 2.23·27-s + 5.91i·29-s + 13.2·33-s − 5.91·35-s − 15.0i·39-s + ⋯
L(s)  = 1  + 1.29·3-s + 0.999i·5-s + 0.999i·7-s + 0.666·9-s + 1.78·11-s − 1.86i·13-s + 1.29i·15-s + 1.92·17-s + 1.29i·21-s − 1.00·25-s − 0.430·27-s + 1.09i·29-s + 2.30·33-s − 0.999·35-s − 2.40i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(17.8864\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (1119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :1/2),\ 0.707 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.247221378\)
\(L(\frac12)\) \(\approx\) \(3.247221378\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.23iT \)
7 \( 1 - 2.64iT \)
good3 \( 1 - 2.23T + 3T^{2} \)
11 \( 1 - 5.91T + 11T^{2} \)
13 \( 1 + 6.70iT - 13T^{2} \)
17 \( 1 - 7.93T + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 5.91iT - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 7.93iT - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 12iT - 71T^{2} \)
73 \( 1 + 10.5T + 73T^{2} \)
79 \( 1 + iT - 79T^{2} \)
83 \( 1 - 8.94T + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + 18.5T + 97T^{2} \)
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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

−9.054586497814248133148861962068, −8.363304995316357553177855217852, −7.74952236711717523753894697047, −6.97001730294721334992375775919, −5.96785716132995411517653404715, −5.36875627571746295019514751953, −3.72063509885539824016954138762, −3.28700543291341579496130000792, −2.62529142033039355000664457893, −1.40633029084067118278971789770, 1.15737014343272202687890176959, 1.81767919423031366124221398371, 3.33534818490040472104395224826, 4.05922632874805162625118141169, 4.47594567930187953495049302687, 5.88403378513276906543247334333, 6.77245225815496850241367131586, 7.60610344126290992685876276359, 8.202229681275867930657656261525, 9.106530822878168836460859036550

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