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Properties

Label 2-99-33.32-c1-0-0
Degree 22
Conductor 9999
Sign 0.394−0.918i0.394 - 0.918i
Analytic cond. 0.7905180.790518
Root an. cond. 0.8891110.889111
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  − 1.73·2-s + 0.999·4-s + 1.41i·5-s + 2.44i·7-s + 1.73·8-s − 2.44i·10-s + (1.73 + 2.82i)11-s + 4.89i·13-s − 4.24i·14-s − 5·16-s − 7.34i·19-s + 1.41i·20-s + (−2.99 − 4.89i)22-s − 2.82i·23-s + 2.99·25-s − 8.48i·26-s + ⋯
L(s)  = 1  − 1.22·2-s + 0.499·4-s + 0.632i·5-s + 0.925i·7-s + 0.612·8-s − 0.774i·10-s + (0.522 + 0.852i)11-s + 1.35i·13-s − 1.13i·14-s − 1.25·16-s − 1.68i·19-s + 0.316i·20-s + (−0.639 − 1.04i)22-s − 0.589i·23-s + 0.599·25-s − 1.66i·26-s + ⋯

Functional equation

Λ(s)=(99s/2 ΓC(s) L(s)=((0.394−0.918i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.394 - 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(99s/2 ΓC(s+1/2) L(s)=((0.394−0.918i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.394 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 9999    =    32⋅113^{2} \cdot 11
Sign: 0.394−0.918i0.394 - 0.918i
Analytic conductor: 0.7905180.790518
Root analytic conductor: 0.8891110.889111
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ99(98,⋅)\chi_{99} (98, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 99, ( :1/2), 0.394−0.918i)(2,\ 99,\ (\ :1/2),\ 0.394 - 0.918i)

Particular Values

L(1)L(1) ≈\approx 0.444946+0.293090i0.444946 + 0.293090i
L(12)L(\frac12) ≈\approx 0.444946+0.293090i0.444946 + 0.293090i
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)
bad3 1 1
11 1+(−1.73−2.82i)T 1 + (-1.73 - 2.82i)T
good2 1+1.73T+2T2 1 + 1.73T + 2T^{2}
5 1−1.41iT−5T2 1 - 1.41iT - 5T^{2}
7 1−2.44iT−7T2 1 - 2.44iT - 7T^{2}
13 1−4.89iT−13T2 1 - 4.89iT - 13T^{2}
17 1+17T2 1 + 17T^{2}
19 1+7.34iT−19T2 1 + 7.34iT - 19T^{2}
23 1+2.82iT−23T2 1 + 2.82iT - 23T^{2}
29 1+6.92T+29T2 1 + 6.92T + 29T^{2}
31 1+4T+31T2 1 + 4T + 31T^{2}
37 1−8T+37T2 1 - 8T + 37T^{2}
41 1−6.92T+41T2 1 - 6.92T + 41T^{2}
43 1−2.44iT−43T2 1 - 2.44iT - 43T^{2}
47 1+2.82iT−47T2 1 + 2.82iT - 47T^{2}
53 1−9.89iT−53T2 1 - 9.89iT - 53T^{2}
59 1+11.3iT−59T2 1 + 11.3iT - 59T^{2}
61 1+4.89iT−61T2 1 + 4.89iT - 61T^{2}
67 1+4T+67T2 1 + 4T + 67T^{2}
71 1+2.82iT−71T2 1 + 2.82iT - 71T^{2}
73 1−73T2 1 - 73T^{2}
79 1+12.2iT−79T2 1 + 12.2iT - 79T^{2}
83 1−13.8T+83T2 1 - 13.8T + 83T^{2}
89 1+7.07iT−89T2 1 + 7.07iT - 89T^{2}
97 1+10T+97T2 1 + 10T + 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

−14.32168190506878386511217532287, −12.95867149640898152762588436825, −11.60951141270858407331611013609, −10.81890712287236672964724032104, −9.293513445353055567601579767448, −9.121893942793375276437921306793, −7.47898061501936816669013541698, −6.55510790933723557155654578383, −4.56921065867110073474598274548, −2.21566271073912486023524403756, 1.03600880771872794815330881490, 3.88675360045157456971255696593, 5.67625452116918334045905826677, 7.46487163945053998074752161708, 8.231727043870037728492129332356, 9.324434967429838102217158277205, 10.32622616838747214066777289003, 11.17186905328940756867896011195, 12.71138264788252203224822483288, 13.60771438649453582831865666787

Graph of the ZZ-function along the critical line