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Properties

Label 2-1040-5.4-c1-0-18
Degree 22
Conductor 10401040
Sign 0.662−0.749i0.662 - 0.749i
Analytic cond. 8.304448.30444
Root an. cond. 2.881742.88174
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.481i·3-s + (1.67 + 1.48i)5-s + 0.806i·7-s + 2.76·9-s + 3.67·11-s − i·13-s + (−0.712 + 0.806i)15-s + 1.35i·17-s − 1.67·19-s − 0.387·21-s − 6.48i·23-s + (0.612 + 4.96i)25-s + 2.77i·27-s − 2.41·29-s + 5.28·31-s + ⋯
L(s)  = 1  + 0.277i·3-s + (0.749 + 0.662i)5-s + 0.304i·7-s + 0.922·9-s + 1.10·11-s − 0.277i·13-s + (−0.184 + 0.208i)15-s + 0.327i·17-s − 0.384·19-s − 0.0846·21-s − 1.35i·23-s + (0.122 + 0.992i)25-s + 0.534i·27-s − 0.449·29-s + 0.949·31-s + ⋯

Functional equation

Λ(s)=(1040s/2 ΓC(s) L(s)=((0.662−0.749i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1040s/2 ΓC(s+1/2) L(s)=((0.662−0.749i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10401040    =    24⋅5⋅132^{4} \cdot 5 \cdot 13
Sign: 0.662−0.749i0.662 - 0.749i
Analytic conductor: 8.304448.30444
Root analytic conductor: 2.881742.88174
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1040(209,⋅)\chi_{1040} (209, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1040, ( :1/2), 0.662−0.749i)(2,\ 1040,\ (\ :1/2),\ 0.662 - 0.749i)

Particular Values

L(1)L(1) ≈\approx 2.0794752582.079475258
L(12)L(\frac12) ≈\approx 2.0794752582.079475258
L(32)L(\frac{3}{2}) 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+(−1.67−1.48i)T 1 + (-1.67 - 1.48i)T
13 1+iT 1 + iT
good3 1−0.481iT−3T2 1 - 0.481iT - 3T^{2}
7 1−0.806iT−7T2 1 - 0.806iT - 7T^{2}
11 1−3.67T+11T2 1 - 3.67T + 11T^{2}
17 1−1.35iT−17T2 1 - 1.35iT - 17T^{2}
19 1+1.67T+19T2 1 + 1.67T + 19T^{2}
23 1+6.48iT−23T2 1 + 6.48iT - 23T^{2}
29 1+2.41T+29T2 1 + 2.41T + 29T^{2}
31 1−5.28T+31T2 1 - 5.28T + 31T^{2}
37 1+3.76iT−37T2 1 + 3.76iT - 37T^{2}
41 1+8.31T+41T2 1 + 8.31T + 41T^{2}
43 1−6.79iT−43T2 1 - 6.79iT - 43T^{2}
47 1−3.19iT−47T2 1 - 3.19iT - 47T^{2}
53 1−5.73iT−53T2 1 - 5.73iT - 53T^{2}
59 1−5.98T+59T2 1 - 5.98T + 59T^{2}
61 1+1.76T+61T2 1 + 1.76T + 61T^{2}
67 1−9.89iT−67T2 1 - 9.89iT - 67T^{2}
71 1+8.56T+71T2 1 + 8.56T + 71T^{2}
73 1+11.7iT−73T2 1 + 11.7iT - 73T^{2}
79 1+2.26T+79T2 1 + 2.26T + 79T^{2}
83 1+3.84iT−83T2 1 + 3.84iT - 83T^{2}
89 1+2.77T+89T2 1 + 2.77T + 89T^{2}
97 1−1.87iT−97T2 1 - 1.87iT - 97T^{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

−10.13379277316794797665365267622, −9.297130199821374525774455545737, −8.579890947106659067012014864418, −7.36942566566821325215453980137, −6.56149523530080531488241383977, −5.96455636262384026404962775689, −4.73193693977314403066290682334, −3.84251925168766387061947782554, −2.64292719887928130446488022466, −1.47100908560065327134911604304, 1.13469122716665794176341484181, 1.99938546319313035491945104617, 3.67265261458423998500899151547, 4.54617287573423746095092440702, 5.51831730946595173743683503384, 6.56600789389474782002588093880, 7.11795810908392735664344655447, 8.241067785365236938699942732725, 9.098969085996931658112988970508, 9.764433887185709165679191095262

Graph of the ZZ-function along the critical line