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Properties

Label 2-1785-5.4-c1-0-49
Degree 22
Conductor 17851785
Sign −0.397−0.917i-0.397 - 0.917i
Analytic cond. 14.253214.2532
Root an. cond. 3.775353.77535
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  + 2.75i·2-s + i·3-s − 5.61·4-s + (−2.05 + 0.889i)5-s − 2.75·6-s + i·7-s − 9.96i·8-s − 9-s + (−2.45 − 5.66i)10-s + 5.12·11-s − 5.61i·12-s − 3.06i·13-s − 2.75·14-s + (−0.889 − 2.05i)15-s + 16.2·16-s + i·17-s + ⋯
L(s)  = 1  + 1.95i·2-s + 0.577i·3-s − 2.80·4-s + (−0.917 + 0.397i)5-s − 1.12·6-s + 0.377i·7-s − 3.52i·8-s − 0.333·9-s + (−0.775 − 1.79i)10-s + 1.54·11-s − 1.62i·12-s − 0.849i·13-s − 0.737·14-s + (−0.229 − 0.529i)15-s + 4.07·16-s + 0.242i·17-s + ⋯

Functional equation

Λ(s)=(1785s/2 ΓC(s) L(s)=((−0.397−0.917i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1785 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.397 - 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1785s/2 ΓC(s+1/2) L(s)=((−0.397−0.917i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1785 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.397 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 17851785    =    3⋅5⋅7⋅173 \cdot 5 \cdot 7 \cdot 17
Sign: −0.397−0.917i-0.397 - 0.917i
Analytic conductor: 14.253214.2532
Root analytic conductor: 3.775353.77535
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1785(1429,⋅)\chi_{1785} (1429, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1785, ( :1/2), −0.397−0.917i)(2,\ 1785,\ (\ :1/2),\ -0.397 - 0.917i)

Particular Values

L(1)L(1) ≈\approx 1.0495931691.049593169
L(12)L(\frac12) ≈\approx 1.0495931691.049593169
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−iT 1 - iT
5 1+(2.05−0.889i)T 1 + (2.05 - 0.889i)T
7 1−iT 1 - iT
17 1−iT 1 - iT
good2 1−2.75iT−2T2 1 - 2.75iT - 2T^{2}
11 1−5.12T+11T2 1 - 5.12T + 11T^{2}
13 1+3.06iT−13T2 1 + 3.06iT - 13T^{2}
19 1−4.71T+19T2 1 - 4.71T + 19T^{2}
23 1+9.27iT−23T2 1 + 9.27iT - 23T^{2}
29 1−0.219T+29T2 1 - 0.219T + 29T^{2}
31 1−1.84T+31T2 1 - 1.84T + 31T^{2}
37 1+9.19iT−37T2 1 + 9.19iT - 37T^{2}
41 1+8.15T+41T2 1 + 8.15T + 41T^{2}
43 1−4.99iT−43T2 1 - 4.99iT - 43T^{2}
47 1+9.93iT−47T2 1 + 9.93iT - 47T^{2}
53 1+2.34iT−53T2 1 + 2.34iT - 53T^{2}
59 1+4.05T+59T2 1 + 4.05T + 59T^{2}
61 1−4.39T+61T2 1 - 4.39T + 61T^{2}
67 1+7.20iT−67T2 1 + 7.20iT - 67T^{2}
71 1+2.47T+71T2 1 + 2.47T + 71T^{2}
73 1−3.44iT−73T2 1 - 3.44iT - 73T^{2}
79 1−11.4T+79T2 1 - 11.4T + 79T^{2}
83 1+8.57iT−83T2 1 + 8.57iT - 83T^{2}
89 1−0.890T+89T2 1 - 0.890T + 89T^{2}
97 1−0.845iT−97T2 1 - 0.845iT - 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

−9.168832164383297242581978916645, −8.493949662478052621655804399457, −7.996321342309425180694870711417, −6.99974489747021083596913314993, −6.50853613458764760614753448691, −5.63378711574212705797694332485, −4.76591998305137046350254865013, −4.01425030343704692991711906992, −3.27975829481015885559435281677, −0.52996619752673975603179502727, 1.08687042286927088933088466555, 1.56440825876175274764540349901, 3.11584586218547652920908827461, 3.73680279227698435194480665039, 4.47714982337864102126765923883, 5.37659661357323649551494784319, 6.79465754974739492846282807167, 7.73635739615574185318189698191, 8.541980651224765921184048424862, 9.366635219797403419037652680796

Graph of the ZZ-function along the critical line