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Properties

Label 2-3276-13.4-c1-0-23
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(2773,⋅)\chi_{3276} (2773, \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

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

Graph of the ZZ-function along the critical line