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Properties

Label 2-3276-13.10-c1-0-18
Degree 22
Conductor 32763276
Sign 0.979+0.202i0.979 + 0.202i
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  + 1.40i·5-s + (−0.866 − 0.5i)7-s + (1.03 − 0.596i)11-s + (−1.17 − 3.40i)13-s + (−2.57 + 4.45i)17-s + (−1.13 − 0.656i)19-s + (−0.660 − 1.14i)23-s + 3.02·25-s + (2.70 + 4.68i)29-s − 7.30i·31-s + (0.702 − 1.21i)35-s + (7.44 − 4.29i)37-s + (2.44 − 1.41i)41-s + (−0.343 + 0.595i)43-s + 1.80i·47-s + ⋯
L(s)  = 1  + 0.628i·5-s + (−0.327 − 0.188i)7-s + (0.311 − 0.179i)11-s + (−0.326 − 0.945i)13-s + (−0.624 + 1.08i)17-s + (−0.260 − 0.150i)19-s + (−0.137 − 0.238i)23-s + 0.604·25-s + (0.501 + 0.869i)29-s − 1.31i·31-s + (0.118 − 0.205i)35-s + (1.22 − 0.706i)37-s + (0.381 − 0.220i)41-s + (−0.0524 + 0.0908i)43-s + 0.263i·47-s + ⋯

Functional equation

Λ(s)=(3276s/2 ΓC(s) L(s)=((0.979+0.202i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.202i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3276s/2 ΓC(s+1/2) L(s)=((0.979+0.202i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 + 0.202i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32763276    =    22⋅32⋅7⋅132^{2} \cdot 3^{2} \cdot 7 \cdot 13
Sign: 0.979+0.202i0.979 + 0.202i
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), 0.979+0.202i)(2,\ 3276,\ (\ :1/2),\ 0.979 + 0.202i)

Particular Values

L(1)L(1) ≈\approx 1.6324401681.632440168
L(12)L(\frac12) ≈\approx 1.6324401681.632440168
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+(1.17+3.40i)T 1 + (1.17 + 3.40i)T
good5 1−1.40iT−5T2 1 - 1.40iT - 5T^{2}
11 1+(−1.03+0.596i)T+(5.5−9.52i)T2 1 + (-1.03 + 0.596i)T + (5.5 - 9.52i)T^{2}
17 1+(2.57−4.45i)T+(−8.5−14.7i)T2 1 + (2.57 - 4.45i)T + (-8.5 - 14.7i)T^{2}
19 1+(1.13+0.656i)T+(9.5+16.4i)T2 1 + (1.13 + 0.656i)T + (9.5 + 16.4i)T^{2}
23 1+(0.660+1.14i)T+(−11.5+19.9i)T2 1 + (0.660 + 1.14i)T + (-11.5 + 19.9i)T^{2}
29 1+(−2.70−4.68i)T+(−14.5+25.1i)T2 1 + (-2.70 - 4.68i)T + (-14.5 + 25.1i)T^{2}
31 1+7.30iT−31T2 1 + 7.30iT - 31T^{2}
37 1+(−7.44+4.29i)T+(18.5−32.0i)T2 1 + (-7.44 + 4.29i)T + (18.5 - 32.0i)T^{2}
41 1+(−2.44+1.41i)T+(20.5−35.5i)T2 1 + (-2.44 + 1.41i)T + (20.5 - 35.5i)T^{2}
43 1+(0.343−0.595i)T+(−21.5−37.2i)T2 1 + (0.343 - 0.595i)T + (-21.5 - 37.2i)T^{2}
47 1−1.80iT−47T2 1 - 1.80iT - 47T^{2}
53 1−6.42T+53T2 1 - 6.42T + 53T^{2}
59 1+(−2.25−1.30i)T+(29.5+51.0i)T2 1 + (-2.25 - 1.30i)T + (29.5 + 51.0i)T^{2}
61 1+(−0.0222+0.0384i)T+(−30.5−52.8i)T2 1 + (-0.0222 + 0.0384i)T + (-30.5 - 52.8i)T^{2}
67 1+(2.14−1.23i)T+(33.5−58.0i)T2 1 + (2.14 - 1.23i)T + (33.5 - 58.0i)T^{2}
71 1+(−2.94−1.70i)T+(35.5+61.4i)T2 1 + (-2.94 - 1.70i)T + (35.5 + 61.4i)T^{2}
73 1+13.2iT−73T2 1 + 13.2iT - 73T^{2}
79 1−7.59T+79T2 1 - 7.59T + 79T^{2}
83 1+8.92iT−83T2 1 + 8.92iT - 83T^{2}
89 1+(5.99−3.45i)T+(44.5−77.0i)T2 1 + (5.99 - 3.45i)T + (44.5 - 77.0i)T^{2}
97 1+(−11.4−6.60i)T+(48.5+84.0i)T2 1 + (-11.4 - 6.60i)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

−8.606180037309770061053118463192, −7.82954448761381091095592795937, −7.11772076767434744026075723163, −6.32295078574522131764073273114, −5.81438308335104434561401050242, −4.68082873666588341491437816972, −3.86857858350648198935449323644, −3.00446825411053676108330810167, −2.16299769136513202674803282273, −0.67558772767604407609006503910, 0.856568848229762355328456114754, 2.08815127034196016917504354922, 2.99262030196989116275496682641, 4.20947854682877491449184565037, 4.71346424313911051548876595710, 5.57189613350865831965933958778, 6.61142310098924026986528921540, 6.97575161295096860848413208151, 8.041631419669559932027910148689, 8.760016907597191260126623343442

Graph of the ZZ-function along the critical line