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Properties

Label 2-2900-5.4-c1-0-16
Degree 22
Conductor 29002900
Sign −0.447−0.894i-0.447 - 0.894i
Analytic cond. 23.156623.1566
Root an. cond. 4.812134.81213
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  + 3.32i·3-s − 1.32i·7-s − 8.02·9-s + 5.32·11-s − 5.02i·13-s + 6.34i·17-s + 4.34·19-s + 4.38·21-s − 1.70i·23-s − 16.6i·27-s + 29-s + 8.34·31-s + 17.6i·33-s + 6.93i·37-s + 16.6·39-s + ⋯
L(s)  = 1  + 1.91i·3-s − 0.499i·7-s − 2.67·9-s + 1.60·11-s − 1.39i·13-s + 1.53i·17-s + 0.997·19-s + 0.957·21-s − 0.356i·23-s − 3.21i·27-s + 0.185·29-s + 1.49·31-s + 3.07i·33-s + 1.14i·37-s + 2.67·39-s + ⋯

Functional equation

Λ(s)=(2900s/2 ΓC(s) L(s)=((−0.447−0.894i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2900s/2 ΓC(s+1/2) L(s)=((−0.447−0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 29002900    =    22⋅52⋅292^{2} \cdot 5^{2} \cdot 29
Sign: −0.447−0.894i-0.447 - 0.894i
Analytic conductor: 23.156623.1566
Root analytic conductor: 4.812134.81213
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2900(349,⋅)\chi_{2900} (349, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2900, ( :1/2), −0.447−0.894i)(2,\ 2900,\ (\ :1/2),\ -0.447 - 0.894i)

Particular Values

L(1)L(1) ≈\approx 1.9079911271.907991127
L(12)L(\frac12) ≈\approx 1.9079911271.907991127
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
29 1−T 1 - T
good3 1−3.32iT−3T2 1 - 3.32iT - 3T^{2}
7 1+1.32iT−7T2 1 + 1.32iT - 7T^{2}
11 1−5.32T+11T2 1 - 5.32T + 11T^{2}
13 1+5.02iT−13T2 1 + 5.02iT - 13T^{2}
17 1−6.34iT−17T2 1 - 6.34iT - 17T^{2}
19 1−4.34T+19T2 1 - 4.34T + 19T^{2}
23 1+1.70iT−23T2 1 + 1.70iT - 23T^{2}
31 1−8.34T+31T2 1 - 8.34T + 31T^{2}
37 1−6.93iT−37T2 1 - 6.93iT - 37T^{2}
41 1+1.02T+41T2 1 + 1.02T + 41T^{2}
43 1−10.7iT−43T2 1 - 10.7iT - 43T^{2}
47 1−0.679iT−47T2 1 - 0.679iT - 47T^{2}
53 1−2.38iT−53T2 1 - 2.38iT - 53T^{2}
59 1+10.4T+59T2 1 + 10.4T + 59T^{2}
61 1+6.38T+61T2 1 + 6.38T + 61T^{2}
67 1−5.70iT−67T2 1 - 5.70iT - 67T^{2}
71 1+3.61T+71T2 1 + 3.61T + 71T^{2}
73 1+6.73iT−73T2 1 + 6.73iT - 73T^{2}
79 1−11.3T+79T2 1 - 11.3T + 79T^{2}
83 1+3.96iT−83T2 1 + 3.96iT - 83T^{2}
89 1+2.58T+89T2 1 + 2.58T + 89T^{2}
97 1−15.3iT−97T2 1 - 15.3iT - 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

−9.168808684696348936758518713465, −8.419290309315023821305993803093, −7.81726887498035459006856975824, −6.35892965880746102328801111308, −5.95006218842075434436456532676, −4.85959900155327462509477696747, −4.31737817361596095833677590014, −3.51865179567782703564229555589, −2.97110453432116412779188712728, −1.08786333741248214862620771551, 0.74714789826024250908497805750, 1.66284136756615919445624543681, 2.46695073368184012527957424173, 3.47286901084952499372442918640, 4.75525054152828781931420743352, 5.78609873738579538250312131314, 6.42316289941697772913270207705, 7.10566472384534556754457525087, 7.42039572754321893419614562833, 8.560198263888960675763607158861

Graph of the ZZ-function along the critical line