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Properties

Label 2-2240-280.139-c1-0-17
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  + 2.23·5-s + 2.64i·7-s − 3·9-s − 5.29·17-s + 4.47i·19-s + 5.00·25-s + 5.91i·35-s − 11.8·37-s + 11.8i·43-s − 6.70·45-s + 5.29i·47-s − 7.00·49-s − 11.8·53-s − 13.4i·59-s + 4.47·61-s + ⋯
L(s)  = 1  + 0.999·5-s + 0.999i·7-s − 9-s − 1.28·17-s + 1.02i·19-s + 1.00·25-s + 0.999i·35-s − 1.94·37-s + 1.80i·43-s − 0.999·45-s + 0.771i·47-s − 49-s − 1.62·53-s − 1.74i·59-s + 0.572·61-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

\(L(1)\) \(\approx\) \(1.063009758\)
\(L(\frac12)\) \(\approx\) \(1.063009758\)
\(L(\frac{3}{2})\) 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 \( 1 + 3T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 5.29T + 17T^{2} \)
19 \( 1 - 4.47iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 11.8T + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 11.8iT - 43T^{2} \)
47 \( 1 - 5.29iT - 47T^{2} \)
53 \( 1 + 11.8T + 53T^{2} \)
59 \( 1 + 13.4iT - 59T^{2} \)
61 \( 1 - 4.47T + 61T^{2} \)
67 \( 1 - 11.8iT - 67T^{2} \)
71 \( 1 - 2iT - 71T^{2} \)
73 \( 1 - 15.8T + 73T^{2} \)
79 \( 1 + 6iT - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + 5.29T + 97T^{2} \)
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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

−9.304736156865456842713888859819, −8.601713506814016208862141720227, −8.062792603386481058507839288936, −6.69245332319294540324547477695, −6.17727214922948711065841902895, −5.45905384206092367533961551325, −4.77186654411510841389194045791, −3.36403046661356669829510772180, −2.49363470595780825362197840657, −1.69153614405403263849130072165, 0.33022158271496790825954400422, 1.82120897769803262164882015354, 2.75727931019421162441684714846, 3.79933313484737454842968909007, 4.89066686626280988828581148978, 5.49362584340279816554895383343, 6.65812749375331241448362734069, 6.85973089239679859053394792827, 8.075893796230036710223878202827, 8.914449069072414487959344888599

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