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Properties

Label 2-3276-13.10-c1-0-15
Degree 22
Conductor 32763276
Sign 0.801−0.598i0.801 - 0.598i
Analytic cond. 26.158926.1589
Root an. cond. 5.114585.11458
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  + 0.118i·5-s + (0.866 + 0.5i)7-s + (3.78 − 2.18i)11-s + (3.33 + 1.38i)13-s + (−3.22 + 5.58i)17-s + (−3.42 − 1.97i)19-s + (−1.34 − 2.32i)23-s + 4.98·25-s + (3.54 + 6.14i)29-s + 8.53i·31-s + (−0.0590 + 0.102i)35-s + (−2.29 + 1.32i)37-s + (1.07 − 0.621i)41-s + (5.45 − 9.45i)43-s + 1.40i·47-s + ⋯
L(s)  = 1  + 0.0528i·5-s + (0.327 + 0.188i)7-s + (1.14 − 0.659i)11-s + (0.923 + 0.382i)13-s + (−0.782 + 1.35i)17-s + (−0.785 − 0.453i)19-s + (−0.280 − 0.485i)23-s + 0.997·25-s + (0.658 + 1.14i)29-s + 1.53i·31-s + (−0.00998 + 0.0172i)35-s + (−0.377 + 0.217i)37-s + (0.168 − 0.0970i)41-s + (0.832 − 1.44i)43-s + 0.204i·47-s + ⋯

Functional equation

Λ(s)=(3276s/2 ΓC(s) L(s)=((0.801−0.598i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 - 0.598i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3276s/2 ΓC(s+1/2) L(s)=((0.801−0.598i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.801 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32763276    =    22⋅32⋅7⋅132^{2} \cdot 3^{2} \cdot 7 \cdot 13
Sign: 0.801−0.598i0.801 - 0.598i
Analytic conductor: 26.158926.1589
Root analytic conductor: 5.114585.11458
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3276(1765,⋅)\chi_{3276} (1765, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3276, ( :1/2), 0.801−0.598i)(2,\ 3276,\ (\ :1/2),\ 0.801 - 0.598i)

Particular Values

L(1)L(1) ≈\approx 2.0866821072.086682107
L(12)L(\frac12) ≈\approx 2.0866821072.086682107
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
3 1 1
7 1+(−0.866−0.5i)T 1 + (-0.866 - 0.5i)T
13 1+(−3.33−1.38i)T 1 + (-3.33 - 1.38i)T
good5 1−0.118iT−5T2 1 - 0.118iT - 5T^{2}
11 1+(−3.78+2.18i)T+(5.5−9.52i)T2 1 + (-3.78 + 2.18i)T + (5.5 - 9.52i)T^{2}
17 1+(3.22−5.58i)T+(−8.5−14.7i)T2 1 + (3.22 - 5.58i)T + (-8.5 - 14.7i)T^{2}
19 1+(3.42+1.97i)T+(9.5+16.4i)T2 1 + (3.42 + 1.97i)T + (9.5 + 16.4i)T^{2}
23 1+(1.34+2.32i)T+(−11.5+19.9i)T2 1 + (1.34 + 2.32i)T + (-11.5 + 19.9i)T^{2}
29 1+(−3.54−6.14i)T+(−14.5+25.1i)T2 1 + (-3.54 - 6.14i)T + (-14.5 + 25.1i)T^{2}
31 1−8.53iT−31T2 1 - 8.53iT - 31T^{2}
37 1+(2.29−1.32i)T+(18.5−32.0i)T2 1 + (2.29 - 1.32i)T + (18.5 - 32.0i)T^{2}
41 1+(−1.07+0.621i)T+(20.5−35.5i)T2 1 + (-1.07 + 0.621i)T + (20.5 - 35.5i)T^{2}
43 1+(−5.45+9.45i)T+(−21.5−37.2i)T2 1 + (-5.45 + 9.45i)T + (-21.5 - 37.2i)T^{2}
47 1−1.40iT−47T2 1 - 1.40iT - 47T^{2}
53 1+5.89T+53T2 1 + 5.89T + 53T^{2}
59 1+(−6.96−4.02i)T+(29.5+51.0i)T2 1 + (-6.96 - 4.02i)T + (29.5 + 51.0i)T^{2}
61 1+(−3.00+5.20i)T+(−30.5−52.8i)T2 1 + (-3.00 + 5.20i)T + (-30.5 - 52.8i)T^{2}
67 1+(−10.6+6.12i)T+(33.5−58.0i)T2 1 + (-10.6 + 6.12i)T + (33.5 - 58.0i)T^{2}
71 1+(10.4+6.05i)T+(35.5+61.4i)T2 1 + (10.4 + 6.05i)T + (35.5 + 61.4i)T^{2}
73 1−4.58iT−73T2 1 - 4.58iT - 73T^{2}
79 1+8.42T+79T2 1 + 8.42T + 79T^{2}
83 1−8.94iT−83T2 1 - 8.94iT - 83T^{2}
89 1+(1.28−0.741i)T+(44.5−77.0i)T2 1 + (1.28 - 0.741i)T + (44.5 - 77.0i)T^{2}
97 1+(0.00126+0.000728i)T+(48.5+84.0i)T2 1 + (0.00126 + 0.000728i)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

−8.631778935667068185517628319400, −8.401196312300489699508088601118, −6.90418901283911191465623821025, −6.59557231220891705982122657900, −5.83902011362928566366055181946, −4.78632928878055958000013145574, −4.02019613427899060218104481548, −3.27540386304074469377522445154, −2.01169256507244949046572542182, −1.10175901830411171702111043059, 0.76056358226341084154694272677, 1.88189190615340209571424287909, 2.90808199611483756642358284528, 4.15988554015732749868985836857, 4.42230807827983426709844116211, 5.61206741748597655696230063536, 6.38059545441856087379212682684, 7.01755845663116900435789872265, 7.86294981939487085307116986987, 8.571813807419987176278513784931

Graph of the ZZ-function along the critical line