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Properties

Label 2-190-5.4-c1-0-1
Degree 22
Conductor 190190
Sign −0.948−0.316i-0.948 - 0.316i
Analytic cond. 1.517151.51715
Root an. cond. 1.231721.23172
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·2-s + 2.41i·3-s − 4-s + (−0.707 + 2.12i)5-s − 2.41·6-s − 1.58i·7-s − i·8-s − 2.82·9-s + (−2.12 − 0.707i)10-s + 1.41·11-s − 2.41i·12-s + 0.171i·13-s + 1.58·14-s + (−5.12 − 1.70i)15-s + 16-s − i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.39i·3-s − 0.5·4-s + (−0.316 + 0.948i)5-s − 0.985·6-s − 0.599i·7-s − 0.353i·8-s − 0.942·9-s + (−0.670 − 0.223i)10-s + 0.426·11-s − 0.696i·12-s + 0.0475i·13-s + 0.423·14-s + (−1.32 − 0.440i)15-s + 0.250·16-s − 0.242i·17-s + ⋯

Functional equation

Λ(s)=(190s/2 ΓC(s) L(s)=((−0.948−0.316i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(190s/2 ΓC(s+1/2) L(s)=((−0.948−0.316i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 190190    =    2⋅5⋅192 \cdot 5 \cdot 19
Sign: −0.948−0.316i-0.948 - 0.316i
Analytic conductor: 1.517151.51715
Root analytic conductor: 1.231721.23172
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ190(39,⋅)\chi_{190} (39, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 190, ( :1/2), −0.948−0.316i)(2,\ 190,\ (\ :1/2),\ -0.948 - 0.316i)

Particular Values

L(1)L(1) ≈\approx 0.165606+1.02051i0.165606 + 1.02051i
L(12)L(\frac12) ≈\approx 0.165606+1.02051i0.165606 + 1.02051i
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−iT 1 - iT
5 1+(0.707−2.12i)T 1 + (0.707 - 2.12i)T
19 1−T 1 - T
good3 1−2.41iT−3T2 1 - 2.41iT - 3T^{2}
7 1+1.58iT−7T2 1 + 1.58iT - 7T^{2}
11 1−1.41T+11T2 1 - 1.41T + 11T^{2}
13 1−0.171iT−13T2 1 - 0.171iT - 13T^{2}
17 1+iT−17T2 1 + iT - 17T^{2}
23 1−9.24iT−23T2 1 - 9.24iT - 23T^{2}
29 1−5.82T+29T2 1 - 5.82T + 29T^{2}
31 1+2.24T+31T2 1 + 2.24T + 31T^{2}
37 1+8.48iT−37T2 1 + 8.48iT - 37T^{2}
41 1−4.24T+41T2 1 - 4.24T + 41T^{2}
43 1−10.2iT−43T2 1 - 10.2iT - 43T^{2}
47 1−47T2 1 - 47T^{2}
53 1+11.4iT−53T2 1 + 11.4iT - 53T^{2}
59 1−12.8T+59T2 1 - 12.8T + 59T^{2}
61 1−5.75T+61T2 1 - 5.75T + 61T^{2}
67 1+13.2iT−67T2 1 + 13.2iT - 67T^{2}
71 1+10.5T+71T2 1 + 10.5T + 71T^{2}
73 1+5.48iT−73T2 1 + 5.48iT - 73T^{2}
79 1+10.4T+79T2 1 + 10.4T + 79T^{2}
83 1+2.48iT−83T2 1 + 2.48iT - 83T^{2}
89 1−7.07T+89T2 1 - 7.07T + 89T^{2}
97 1−11.6iT−97T2 1 - 11.6iT - 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

−13.34346586336105038487837998677, −11.69604908165872583763493940479, −10.84901936046373798613611491902, −9.944956264859400231546112847225, −9.226144204304388266739620464857, −7.79979579212299474698012779559, −6.86047977285913236053756656268, −5.52828841874755400543496406976, −4.23857975780464608175410516167, −3.40229962401988518157623825290, 1.02863569598538131335492315274, 2.47081895956652662150183068552, 4.34788560192156619957240912078, 5.75494901080164014983248105951, 7.00122589752917130978013107794, 8.331790816653528889246242339463, 8.795189014303045954545592437461, 10.22131867102649119884809628737, 11.65936551462715246284612406980, 12.21325008976842428965606723517

Graph of the ZZ-function along the critical line