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Properties

Label 2-1560-5.4-c1-0-10
Degree 22
Conductor 15601560
Sign 0.807−0.590i0.807 - 0.590i
Analytic cond. 12.456612.4566
Root an. cond. 3.529393.52939
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  − i·3-s + (1.32 + 1.80i)5-s − 9-s − 4.64·11-s − i·13-s + (1.80 − 1.32i)15-s + 4.24i·17-s + 6.24·19-s − 2.24i·23-s + (−1.51 + 4.76i)25-s + i·27-s + 9.21·29-s + 9.28·31-s + 4.64i·33-s + 7.28i·37-s + ⋯
L(s)  = 1  − 0.577i·3-s + (0.590 + 0.807i)5-s − 0.333·9-s − 1.39·11-s − 0.277i·13-s + (0.466 − 0.340i)15-s + 1.03i·17-s + 1.43·19-s − 0.469i·23-s + (−0.303 + 0.952i)25-s + 0.192i·27-s + 1.71·29-s + 1.66·31-s + 0.807i·33-s + 1.19i·37-s + ⋯

Functional equation

Λ(s)=(1560s/2 ΓC(s) L(s)=((0.807−0.590i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.807 - 0.590i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1560s/2 ΓC(s+1/2) L(s)=((0.807−0.590i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.807 - 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 15601560    =    23⋅3⋅5⋅132^{3} \cdot 3 \cdot 5 \cdot 13
Sign: 0.807−0.590i0.807 - 0.590i
Analytic conductor: 12.456612.4566
Root analytic conductor: 3.529393.52939
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1560(1249,⋅)\chi_{1560} (1249, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1560, ( :1/2), 0.807−0.590i)(2,\ 1560,\ (\ :1/2),\ 0.807 - 0.590i)

Particular Values

L(1)L(1) ≈\approx 1.6911658381.691165838
L(12)L(\frac12) ≈\approx 1.6911658381.691165838
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
3 1+iT 1 + iT
5 1+(−1.32−1.80i)T 1 + (-1.32 - 1.80i)T
13 1+iT 1 + iT
good7 1−7T2 1 - 7T^{2}
11 1+4.64T+11T2 1 + 4.64T + 11T^{2}
17 1−4.24iT−17T2 1 - 4.24iT - 17T^{2}
19 1−6.24T+19T2 1 - 6.24T + 19T^{2}
23 1+2.24iT−23T2 1 + 2.24iT - 23T^{2}
29 1−9.21T+29T2 1 - 9.21T + 29T^{2}
31 1−9.28T+31T2 1 - 9.28T + 31T^{2}
37 1−7.28iT−37T2 1 - 7.28iT - 37T^{2}
41 1+5.67T+41T2 1 + 5.67T + 41T^{2}
43 1−4.24iT−43T2 1 - 4.24iT - 43T^{2}
47 1−2.88iT−47T2 1 - 2.88iT - 47T^{2}
53 1−9.21iT−53T2 1 - 9.21iT - 53T^{2}
59 1+5.92T+59T2 1 + 5.92T + 59T^{2}
61 1−0.969T+61T2 1 - 0.969T + 61T^{2}
67 1+1.93iT−67T2 1 + 1.93iT - 67T^{2}
71 1−5.60T+71T2 1 - 5.60T + 71T^{2}
73 1−12.5iT−73T2 1 - 12.5iT - 73T^{2}
79 1+12.2T+79T2 1 + 12.2T + 79T^{2}
83 1−3.67iT−83T2 1 - 3.67iT - 83T^{2}
89 1−9.67T+89T2 1 - 9.67T + 89T^{2}
97 1+6iT−97T2 1 + 6iT - 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

−9.778573235752966630884217695802, −8.441942384824842617661871719080, −7.964163989842803213976498035922, −7.06639634395306032651721100231, −6.30274074092437739551701187249, −5.59357451925045113239189400816, −4.64982929997061365432111977485, −3.05450190799556185236435560736, −2.64252810369303682756033193900, −1.23068546542277125054153530126, 0.73764770233626992639436083242, 2.32965637623613612762920136390, 3.23531791428257812801357020721, 4.63183174145057538038395255810, 5.11171818288867438888292609926, 5.80774251948832226749467556079, 6.97819884587705216246658747807, 7.927900575240723868640596805603, 8.635013415150993266327540199425, 9.480714188220271973593980236037

Graph of the ZZ-function along the critical line