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Properties

Label 2-2900-5.4-c1-0-10
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  + 1.35i·3-s + 0.648i·7-s + 1.17·9-s + 3.35·11-s + 4.17i·13-s − 4.82i·17-s − 6.82·19-s − 0.876·21-s + 5.52i·23-s + 5.64i·27-s + 29-s − 2.82·31-s + 4.53i·33-s + 10.2i·37-s − 5.64·39-s + ⋯
L(s)  = 1  + 0.780i·3-s + 0.244i·7-s + 0.390·9-s + 1.01·11-s + 1.15i·13-s − 1.16i·17-s − 1.56·19-s − 0.191·21-s + 1.15i·23-s + 1.08i·27-s + 0.185·29-s − 0.506·31-s + 0.788i·33-s + 1.68i·37-s − 0.903·39-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 1.7433069861.743306986
L(12)L(\frac12) ≈\approx 1.7433069861.743306986
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 1−1.35iT−3T2 1 - 1.35iT - 3T^{2}
7 1−0.648iT−7T2 1 - 0.648iT - 7T^{2}
11 1−3.35T+11T2 1 - 3.35T + 11T^{2}
13 1−4.17iT−13T2 1 - 4.17iT - 13T^{2}
17 1+4.82iT−17T2 1 + 4.82iT - 17T^{2}
19 1+6.82T+19T2 1 + 6.82T + 19T^{2}
23 1−5.52iT−23T2 1 - 5.52iT - 23T^{2}
31 1+2.82T+31T2 1 + 2.82T + 31T^{2}
37 1−10.2iT−37T2 1 - 10.2iT - 37T^{2}
41 1−8.17T+41T2 1 - 8.17T + 41T^{2}
43 1+5.69iT−43T2 1 + 5.69iT - 43T^{2}
47 1−2.64iT−47T2 1 - 2.64iT - 47T^{2}
53 1+2.87iT−53T2 1 + 2.87iT - 53T^{2}
59 1−13.2T+59T2 1 - 13.2T + 59T^{2}
61 1+1.12T+61T2 1 + 1.12T + 61T^{2}
67 1+1.52iT−67T2 1 + 1.52iT - 67T^{2}
71 1+8.87T+71T2 1 + 8.87T + 71T^{2}
73 1−9.69iT−73T2 1 - 9.69iT - 73T^{2}
79 1+8.99T+79T2 1 + 8.99T + 79T^{2}
83 1−1.94iT−83T2 1 - 1.94iT - 83T^{2}
89 1+17.0T+89T2 1 + 17.0T + 89T^{2}
97 1−13.3iT−97T2 1 - 13.3iT - 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

−9.063119354427458163875541908440, −8.581473287291675528289791392970, −7.29974177682906855388850052778, −6.82982004722803072495128277070, −5.95029195213226924727302777228, −4.95118582881574650965086419213, −4.24674496459150001812706178548, −3.70056172585261031554346655812, −2.43302752090254754718850329228, −1.35396691623909700858900142909, 0.58194129699924828455124599468, 1.66416224237692529743728135055, 2.57693226430316908732589183835, 3.94532081001897662350759861983, 4.33055597049063262116611691155, 5.74895585758420298926867317282, 6.28492155182959352457335551343, 7.00894716050268245326537207728, 7.72371142317507397484942783856, 8.468520773843406042105779271291

Graph of the ZZ-function along the critical line