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Properties

Label 2-832-13.4-c1-0-11
Degree 22
Conductor 832832
Sign −0.252−0.967i-0.252 - 0.967i
Analytic cond. 6.643556.64355
Root an. cond. 2.577502.57750
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  + (−0.866 + 1.5i)3-s + 3.46i·5-s + (2.59 − 1.5i)7-s + (4.33 + 2.5i)11-s + (1 − 3.46i)13-s + (−5.19 − 2.99i)15-s + (3.5 + 6.06i)17-s + (4.33 − 2.5i)19-s + 5.19i·21-s + (2.59 − 4.5i)23-s − 6.99·25-s − 5.19·27-s + (−2.5 + 4.33i)29-s − 2i·31-s + (−7.5 + 4.33i)33-s + ⋯
L(s)  = 1  + (−0.499 + 0.866i)3-s + 1.54i·5-s + (0.981 − 0.566i)7-s + (1.30 + 0.753i)11-s + (0.277 − 0.960i)13-s + (−1.34 − 0.774i)15-s + (0.848 + 1.47i)17-s + (0.993 − 0.573i)19-s + 1.13i·21-s + (0.541 − 0.938i)23-s − 1.39·25-s − 1.00·27-s + (−0.464 + 0.804i)29-s − 0.359i·31-s + (−1.30 + 0.753i)33-s + ⋯

Functional equation

Λ(s)=(832s/2 ΓC(s) L(s)=((−0.252−0.967i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.252 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(832s/2 ΓC(s+1/2) L(s)=((−0.252−0.967i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 832832    =    26⋅132^{6} \cdot 13
Sign: −0.252−0.967i-0.252 - 0.967i
Analytic conductor: 6.643556.64355
Root analytic conductor: 2.577502.57750
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ832(641,⋅)\chi_{832} (641, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 832, ( :1/2), −0.252−0.967i)(2,\ 832,\ (\ :1/2),\ -0.252 - 0.967i)

Particular Values

L(1)L(1) ≈\approx 1.01275+1.31112i1.01275 + 1.31112i
L(12)L(\frac12) ≈\approx 1.01275+1.31112i1.01275 + 1.31112i
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 1
13 1+(−1+3.46i)T 1 + (-1 + 3.46i)T
good3 1+(0.866−1.5i)T+(−1.5−2.59i)T2 1 + (0.866 - 1.5i)T + (-1.5 - 2.59i)T^{2}
5 1−3.46iT−5T2 1 - 3.46iT - 5T^{2}
7 1+(−2.59+1.5i)T+(3.5−6.06i)T2 1 + (-2.59 + 1.5i)T + (3.5 - 6.06i)T^{2}
11 1+(−4.33−2.5i)T+(5.5+9.52i)T2 1 + (-4.33 - 2.5i)T + (5.5 + 9.52i)T^{2}
17 1+(−3.5−6.06i)T+(−8.5+14.7i)T2 1 + (-3.5 - 6.06i)T + (-8.5 + 14.7i)T^{2}
19 1+(−4.33+2.5i)T+(9.5−16.4i)T2 1 + (-4.33 + 2.5i)T + (9.5 - 16.4i)T^{2}
23 1+(−2.59+4.5i)T+(−11.5−19.9i)T2 1 + (-2.59 + 4.5i)T + (-11.5 - 19.9i)T^{2}
29 1+(2.5−4.33i)T+(−14.5−25.1i)T2 1 + (2.5 - 4.33i)T + (-14.5 - 25.1i)T^{2}
31 1+2iT−31T2 1 + 2iT - 31T^{2}
37 1+(4.5+2.59i)T+(18.5+32.0i)T2 1 + (4.5 + 2.59i)T + (18.5 + 32.0i)T^{2}
41 1+(1.5+0.866i)T+(20.5+35.5i)T2 1 + (1.5 + 0.866i)T + (20.5 + 35.5i)T^{2}
43 1+(2.59+4.5i)T+(−21.5+37.2i)T2 1 + (2.59 + 4.5i)T + (-21.5 + 37.2i)T^{2}
47 1+4iT−47T2 1 + 4iT - 47T^{2}
53 1+4T+53T2 1 + 4T + 53T^{2}
59 1+(6.06−3.5i)T+(29.5−51.0i)T2 1 + (6.06 - 3.5i)T + (29.5 - 51.0i)T^{2}
61 1+(−1.5−2.59i)T+(−30.5+52.8i)T2 1 + (-1.5 - 2.59i)T + (-30.5 + 52.8i)T^{2}
67 1+(−2.59−1.5i)T+(33.5+58.0i)T2 1 + (-2.59 - 1.5i)T + (33.5 + 58.0i)T^{2}
71 1+(6.06−3.5i)T+(35.5−61.4i)T2 1 + (6.06 - 3.5i)T + (35.5 - 61.4i)T^{2}
73 1+3.46iT−73T2 1 + 3.46iT - 73T^{2}
79 1+3.46T+79T2 1 + 3.46T + 79T^{2}
83 1+14iT−83T2 1 + 14iT - 83T^{2}
89 1+(−1.5−0.866i)T+(44.5+77.0i)T2 1 + (-1.5 - 0.866i)T + (44.5 + 77.0i)T^{2}
97 1+(−7.5+4.33i)T+(48.5−84.0i)T2 1 + (-7.5 + 4.33i)T + (48.5 - 84.0i)T^{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

−10.37855538303046272351632864330, −10.14697976683077386098533373380, −8.875654501507909549055467828457, −7.66284171828866836188876963342, −7.10207356397291731865845986530, −6.07121318228199451308780861511, −5.07820939062462916332348453283, −4.05599997123739660320989240945, −3.29395637560105612513211680269, −1.63865794362224080110720347500, 1.09493626110578598642653348604, 1.54146021753320441311826664331, 3.60662304927020795835689457928, 4.85392431774195462267657271737, 5.48757560664145178406598866342, 6.40409613211875862825593521930, 7.48304179116093703173315492948, 8.248679383802748163259605513561, 9.269785943212156608188685962966, 9.441302178174483087967774271641

Graph of the ZZ-function along the critical line