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Properties

Label 2-2240-280.69-c0-0-9
Degree 22
Conductor 22402240
Sign 0.258+0.965i0.258 + 0.965i
Analytic cond. 1.117901.11790
Root an. cond. 1.057311.05731
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73i·3-s − i·5-s − 7-s − 1.99·9-s − i·11-s − i·13-s + 1.73·15-s − 1.73·17-s − 1.73i·21-s − 25-s − 1.73i·27-s − 1.73i·29-s + 1.73·33-s + i·35-s + 1.73·39-s + ⋯
L(s)  = 1  + 1.73i·3-s − i·5-s − 7-s − 1.99·9-s − i·11-s − i·13-s + 1.73·15-s − 1.73·17-s − 1.73i·21-s − 25-s − 1.73i·27-s − 1.73i·29-s + 1.73·33-s + i·35-s + 1.73·39-s + ⋯

Functional equation

Λ(s)=(2240s/2 ΓC(s) L(s)=((0.258+0.965i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2240s/2 ΓC(s) L(s)=((0.258+0.965i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 22402240    =    26⋅5⋅72^{6} \cdot 5 \cdot 7
Sign: 0.258+0.965i0.258 + 0.965i
Analytic conductor: 1.117901.11790
Root analytic conductor: 1.057311.05731
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2240(1889,⋅)\chi_{2240} (1889, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2240, ( :0), 0.258+0.965i)(2,\ 2240,\ (\ :0),\ 0.258 + 0.965i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.48739493850.4873949385
L(12)L(\frac12) ≈\approx 0.48739493850.4873949385
L(1)L(1) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
5 1+iT 1 + iT
7 1+T 1 + T
good3 1−1.73iT−T2 1 - 1.73iT - T^{2}
11 1+iT−T2 1 + iT - T^{2}
13 1+iT−T2 1 + iT - T^{2}
17 1+1.73T+T2 1 + 1.73T + T^{2}
19 1+T2 1 + T^{2}
23 1−T2 1 - T^{2}
29 1+1.73iT−T2 1 + 1.73iT - T^{2}
31 1−T2 1 - T^{2}
37 1+T2 1 + T^{2}
41 1−T2 1 - T^{2}
43 1+T2 1 + T^{2}
47 1+T+T2 1 + T + T^{2}
53 1+T2 1 + T^{2}
59 1+T2 1 + T^{2}
61 1+T2 1 + T^{2}
67 1+T2 1 + T^{2}
71 1+T2 1 + T^{2}
73 1+T2 1 + T^{2}
79 1+1.73T+T2 1 + 1.73T + T^{2}
83 1−T2 1 - T^{2}
89 1−T2 1 - T^{2}
97 1−1.73T+T2 1 - 1.73T + T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−8.996296567814500332823160041875, −8.728261001066961559280491741608, −7.84255267468644537602463459269, −6.33964658938566057641234344380, −5.78485833738005751224287080363, −4.94617531358602030327628906473, −4.20238668693489551237951595240, −3.50989590661150073897940972887, −2.57777634615938663534330396823, −0.31024107971170636968755282649, 1.74609745618083056380768413951, 2.40033494539655666440454706602, 3.33178308771758188666261494745, 4.55538815216758117670035557946, 5.92053155975828033261153612143, 6.67442554826816721663465208511, 6.91075002209753406866659732517, 7.42391635808013777773746707013, 8.564647369609327433872337119672, 9.226163090075877818749898909478

Graph of the ZZ-function along the critical line