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Properties

Label 2-3276-13.10-c1-0-11
Degree 22
Conductor 32763276
Sign 0.861−0.507i0.861 - 0.507i
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  − 2.36i·5-s + (−0.866 − 0.5i)7-s + (−2.56 + 1.47i)11-s + (3.15 + 1.73i)13-s + (−1.25 + 2.17i)17-s + (3.52 + 2.03i)19-s + (1.45 + 2.52i)23-s − 0.583·25-s + (0.464 + 0.805i)29-s − 2.20i·31-s + (−1.18 + 2.04i)35-s + (−5.73 + 3.31i)37-s + (−9.87 + 5.69i)41-s + (0.859 − 1.48i)43-s − 2.64i·47-s + ⋯
L(s)  = 1  − 1.05i·5-s + (−0.327 − 0.188i)7-s + (−0.772 + 0.445i)11-s + (0.876 + 0.481i)13-s + (−0.303 + 0.526i)17-s + (0.808 + 0.466i)19-s + (0.304 + 0.526i)23-s − 0.116·25-s + (0.0863 + 0.149i)29-s − 0.396i·31-s + (−0.199 + 0.345i)35-s + (−0.942 + 0.544i)37-s + (−1.54 + 0.890i)41-s + (0.131 − 0.226i)43-s − 0.385i·47-s + ⋯

Functional equation

Λ(s)=(3276s/2 ΓC(s) L(s)=((0.861−0.507i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.861 - 0.507i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3276s/2 ΓC(s+1/2) L(s)=((0.861−0.507i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.861 - 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32763276    =    22⋅32⋅7⋅132^{2} \cdot 3^{2} \cdot 7 \cdot 13
Sign: 0.861−0.507i0.861 - 0.507i
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), 0.861−0.507i)(2,\ 3276,\ (\ :1/2),\ 0.861 - 0.507i)

Particular Values

L(1)L(1) ≈\approx 1.5176029421.517602942
L(12)L(\frac12) ≈\approx 1.5176029421.517602942
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+(−3.15−1.73i)T 1 + (-3.15 - 1.73i)T
good5 1+2.36iT−5T2 1 + 2.36iT - 5T^{2}
11 1+(2.56−1.47i)T+(5.5−9.52i)T2 1 + (2.56 - 1.47i)T + (5.5 - 9.52i)T^{2}
17 1+(1.25−2.17i)T+(−8.5−14.7i)T2 1 + (1.25 - 2.17i)T + (-8.5 - 14.7i)T^{2}
19 1+(−3.52−2.03i)T+(9.5+16.4i)T2 1 + (-3.52 - 2.03i)T + (9.5 + 16.4i)T^{2}
23 1+(−1.45−2.52i)T+(−11.5+19.9i)T2 1 + (-1.45 - 2.52i)T + (-11.5 + 19.9i)T^{2}
29 1+(−0.464−0.805i)T+(−14.5+25.1i)T2 1 + (-0.464 - 0.805i)T + (-14.5 + 25.1i)T^{2}
31 1+2.20iT−31T2 1 + 2.20iT - 31T^{2}
37 1+(5.73−3.31i)T+(18.5−32.0i)T2 1 + (5.73 - 3.31i)T + (18.5 - 32.0i)T^{2}
41 1+(9.87−5.69i)T+(20.5−35.5i)T2 1 + (9.87 - 5.69i)T + (20.5 - 35.5i)T^{2}
43 1+(−0.859+1.48i)T+(−21.5−37.2i)T2 1 + (-0.859 + 1.48i)T + (-21.5 - 37.2i)T^{2}
47 1+2.64iT−47T2 1 + 2.64iT - 47T^{2}
53 1−0.0492T+53T2 1 - 0.0492T + 53T^{2}
59 1+(−7.11−4.11i)T+(29.5+51.0i)T2 1 + (-7.11 - 4.11i)T + (29.5 + 51.0i)T^{2}
61 1+(1.48−2.57i)T+(−30.5−52.8i)T2 1 + (1.48 - 2.57i)T + (-30.5 - 52.8i)T^{2}
67 1+(−5.51+3.18i)T+(33.5−58.0i)T2 1 + (-5.51 + 3.18i)T + (33.5 - 58.0i)T^{2}
71 1+(−7.58−4.38i)T+(35.5+61.4i)T2 1 + (-7.58 - 4.38i)T + (35.5 + 61.4i)T^{2}
73 1−5.56iT−73T2 1 - 5.56iT - 73T^{2}
79 1−13.6T+79T2 1 - 13.6T + 79T^{2}
83 1−14.7iT−83T2 1 - 14.7iT - 83T^{2}
89 1+(5.06−2.92i)T+(44.5−77.0i)T2 1 + (5.06 - 2.92i)T + (44.5 - 77.0i)T^{2}
97 1+(−14.5−8.42i)T+(48.5+84.0i)T2 1 + (-14.5 - 8.42i)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

−8.550589451823677279469312517676, −8.207712035068102579951249892122, −7.20139856642729197534069674366, −6.49808088970937589491604123306, −5.49336637745443433660015670123, −4.98412104189843539584771123524, −4.04032721904989294427985219578, −3.26677237929679349368457075426, −1.94158539861616779490488882816, −1.00841404274724712868692172284, 0.55078653879224086544511444314, 2.16908553815286596903041309933, 3.11477987560096763830107778213, 3.49359905914387186004153597705, 4.89570015167530618405313347494, 5.55939198821869460888492738943, 6.47368791414788409439881754575, 6.97812710984447240361241860628, 7.78992021168815372110939871327, 8.591469207264023184291980220844

Graph of the ZZ-function along the critical line