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Properties

Label 2-1805-5.4-c1-0-78
Degree 22
Conductor 18051805
Sign 0.839+0.543i0.839 + 0.543i
Analytic cond. 14.412914.4129
Root an. cond. 3.796443.79644
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  − 0.449i·2-s − 1.95i·3-s + 1.79·4-s + (1.87 + 1.21i)5-s − 0.879·6-s + 2.06i·7-s − 1.70i·8-s − 0.829·9-s + (0.545 − 0.843i)10-s − 2.30·11-s − 3.51i·12-s + 4.39i·13-s + 0.926·14-s + (2.37 − 3.67i)15-s + 2.82·16-s + 5.47i·17-s + ⋯
L(s)  = 1  − 0.317i·2-s − 1.12i·3-s + 0.899·4-s + (0.839 + 0.543i)5-s − 0.358·6-s + 0.778i·7-s − 0.603i·8-s − 0.276·9-s + (0.172 − 0.266i)10-s − 0.695·11-s − 1.01i·12-s + 1.21i·13-s + 0.247·14-s + (0.613 − 0.948i)15-s + 0.707·16-s + 1.32i·17-s + ⋯

Functional equation

Λ(s)=(1805s/2 ΓC(s) L(s)=((0.839+0.543i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.839 + 0.543i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1805s/2 ΓC(s+1/2) L(s)=((0.839+0.543i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.839 + 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18051805    =    5⋅1925 \cdot 19^{2}
Sign: 0.839+0.543i0.839 + 0.543i
Analytic conductor: 14.412914.4129
Root analytic conductor: 3.796443.79644
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1805(1084,⋅)\chi_{1805} (1084, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1805, ( :1/2), 0.839+0.543i)(2,\ 1805,\ (\ :1/2),\ 0.839 + 0.543i)

Particular Values

L(1)L(1) ≈\approx 2.6300697532.630069753
L(12)L(\frac12) ≈\approx 2.6300697532.630069753
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)
bad5 1+(−1.87−1.21i)T 1 + (-1.87 - 1.21i)T
19 1 1
good2 1+0.449iT−2T2 1 + 0.449iT - 2T^{2}
3 1+1.95iT−3T2 1 + 1.95iT - 3T^{2}
7 1−2.06iT−7T2 1 - 2.06iT - 7T^{2}
11 1+2.30T+11T2 1 + 2.30T + 11T^{2}
13 1−4.39iT−13T2 1 - 4.39iT - 13T^{2}
17 1−5.47iT−17T2 1 - 5.47iT - 17T^{2}
23 1+5.81iT−23T2 1 + 5.81iT - 23T^{2}
29 1−5.50T+29T2 1 - 5.50T + 29T^{2}
31 1+0.757T+31T2 1 + 0.757T + 31T^{2}
37 1+6.22iT−37T2 1 + 6.22iT - 37T^{2}
41 1−6.53T+41T2 1 - 6.53T + 41T^{2}
43 1−3.16iT−43T2 1 - 3.16iT - 43T^{2}
47 1+6.36iT−47T2 1 + 6.36iT - 47T^{2}
53 1−3.85iT−53T2 1 - 3.85iT - 53T^{2}
59 1−2.55T+59T2 1 - 2.55T + 59T^{2}
61 1−9.94T+61T2 1 - 9.94T + 61T^{2}
67 1−1.70iT−67T2 1 - 1.70iT - 67T^{2}
71 1+9.85T+71T2 1 + 9.85T + 71T^{2}
73 1−10.2iT−73T2 1 - 10.2iT - 73T^{2}
79 1−2.41T+79T2 1 - 2.41T + 79T^{2}
83 1+7.06iT−83T2 1 + 7.06iT - 83T^{2}
89 1+2.33T+89T2 1 + 2.33T + 89T^{2}
97 1−6.81iT−97T2 1 - 6.81iT - 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

−9.176661138551404341515304369136, −8.321988744296589455923967172773, −7.44213799053998538037632301737, −6.65011012284975674898443096056, −6.30201534916654976405125666447, −5.51569922003310481929991357822, −4.04459984865588512970459596904, −2.53615539161218854254440294910, −2.30388385068419052047526844303, −1.34770632930046797494439591199, 1.07334531821126731106111073758, 2.56466478113519372989509832854, 3.37566287188269731330912884920, 4.66890752998936663231482151664, 5.25772691787317753698416798715, 5.92774632081883711542355952488, 7.05537372125825525813073410959, 7.68576148846580952938820170203, 8.574233933266098467875372118539, 9.628198015440086909496145299424

Graph of the ZZ-function along the critical line