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Properties

Label 2-980-28.27-c1-0-58
Degree 22
Conductor 980980
Sign 0.753+0.657i0.753 + 0.657i
Analytic cond. 7.825337.82533
Root an. cond. 2.797382.79738
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  + (0.0982 + 1.41i)2-s + 0.662·3-s + (−1.98 + 0.277i)4-s + i·5-s + (0.0650 + 0.934i)6-s + (−0.585 − 2.76i)8-s − 2.56·9-s + (−1.41 + 0.0982i)10-s − 3.61i·11-s + (−1.31 + 0.183i)12-s − 5.83i·13-s + 0.662i·15-s + (3.84 − 1.09i)16-s − 1.36i·17-s + (−0.251 − 3.61i)18-s − 4.09·19-s + ⋯
L(s)  = 1  + (0.0694 + 0.997i)2-s + 0.382·3-s + (−0.990 + 0.138i)4-s + 0.447i·5-s + (0.0265 + 0.381i)6-s + (−0.207 − 0.978i)8-s − 0.853·9-s + (−0.446 + 0.0310i)10-s − 1.08i·11-s + (−0.378 + 0.0530i)12-s − 1.61i·13-s + 0.171i·15-s + (0.961 − 0.274i)16-s − 0.331i·17-s + (−0.0593 − 0.851i)18-s − 0.939·19-s + ⋯

Functional equation

Λ(s)=(980s/2 ΓC(s) L(s)=((0.753+0.657i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.753 + 0.657i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(980s/2 ΓC(s+1/2) L(s)=((0.753+0.657i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.753 + 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 980980    =    22⋅5⋅722^{2} \cdot 5 \cdot 7^{2}
Sign: 0.753+0.657i0.753 + 0.657i
Analytic conductor: 7.825337.82533
Root analytic conductor: 2.797382.79738
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ980(391,⋅)\chi_{980} (391, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 980, ( :1/2), 0.753+0.657i)(2,\ 980,\ (\ :1/2),\ 0.753 + 0.657i)

Particular Values

L(1)L(1) ≈\approx 0.864521−0.324417i0.864521 - 0.324417i
L(12)L(\frac12) ≈\approx 0.864521−0.324417i0.864521 - 0.324417i
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+(−0.0982−1.41i)T 1 + (-0.0982 - 1.41i)T
5 1−iT 1 - iT
7 1 1
good3 1−0.662T+3T2 1 - 0.662T + 3T^{2}
11 1+3.61iT−11T2 1 + 3.61iT - 11T^{2}
13 1+5.83iT−13T2 1 + 5.83iT - 13T^{2}
17 1+1.36iT−17T2 1 + 1.36iT - 17T^{2}
19 1+4.09T+19T2 1 + 4.09T + 19T^{2}
23 1+3.24iT−23T2 1 + 3.24iT - 23T^{2}
29 1−5.19T+29T2 1 - 5.19T + 29T^{2}
31 1+8.86T+31T2 1 + 8.86T + 31T^{2}
37 1−10.7T+37T2 1 - 10.7T + 37T^{2}
41 1−0.832iT−41T2 1 - 0.832iT - 41T^{2}
43 1−3.10iT−43T2 1 - 3.10iT - 43T^{2}
47 1+6.89T+47T2 1 + 6.89T + 47T^{2}
53 1+7.41T+53T2 1 + 7.41T + 53T^{2}
59 1−7.47T+59T2 1 - 7.47T + 59T^{2}
61 1+1.48iT−61T2 1 + 1.48iT - 61T^{2}
67 1+2.53iT−67T2 1 + 2.53iT - 67T^{2}
71 1+3.52iT−71T2 1 + 3.52iT - 71T^{2}
73 1+5.16iT−73T2 1 + 5.16iT - 73T^{2}
79 1+11.3iT−79T2 1 + 11.3iT - 79T^{2}
83 1+6.49T+83T2 1 + 6.49T + 83T^{2}
89 1+9.39iT−89T2 1 + 9.39iT - 89T^{2}
97 1−0.343iT−97T2 1 - 0.343iT - 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

−9.717688034369685823305318171171, −8.721828084573625508373196014596, −8.216436227895960406437928796520, −7.54733335521393722227974163326, −6.29663833204245101820586208191, −5.86373655145194872235082996084, −4.83290669691920308001160929796, −3.50227054285996059179940298766, −2.80629527809289081558857306607, −0.39056176151843133378241456168, 1.66490217532702041744037573131, 2.47998723324538917069583148863, 3.85266811484609782037519384387, 4.50629766151454209041521897525, 5.53093503210503769265152016513, 6.68022048338221195889249805920, 7.903437322899075400436857862035, 8.721529945988737530962508810220, 9.335286087017120218522891039203, 9.948085475115944419044407743682

Graph of the ZZ-function along the critical line