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Properties

Label 2-2640-12.11-c1-0-14
Degree 22
Conductor 26402640
Sign −0.934+0.356i-0.934 + 0.356i
Analytic cond. 21.080521.0805
Root an. cond. 4.591354.59135
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  + (0.618 + 1.61i)3-s + i·5-s + 4.20i·7-s + (−2.23 + 2.00i)9-s − 11-s + 1.36·13-s + (−1.61 + 0.618i)15-s − 2.20i·17-s + 1.62i·19-s + (−6.80 + 2.60i)21-s − 5.20·23-s − 25-s + (−4.61 − 2.38i)27-s + 0.371i·29-s + 7.20i·31-s + ⋯
L(s)  = 1  + (0.356 + 0.934i)3-s + 0.447i·5-s + 1.59i·7-s + (−0.745 + 0.666i)9-s − 0.301·11-s + 0.378·13-s + (−0.417 + 0.159i)15-s − 0.535i·17-s + 0.373i·19-s + (−1.48 + 0.567i)21-s − 1.08·23-s − 0.200·25-s + (−0.888 − 0.458i)27-s + 0.0689i·29-s + 1.29i·31-s + ⋯

Functional equation

Λ(s)=(2640s/2 ΓC(s) L(s)=((−0.934+0.356i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 + 0.356i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2640s/2 ΓC(s+1/2) L(s)=((−0.934+0.356i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.934 + 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 26402640    =    24⋅3⋅5⋅112^{4} \cdot 3 \cdot 5 \cdot 11
Sign: −0.934+0.356i-0.934 + 0.356i
Analytic conductor: 21.080521.0805
Root analytic conductor: 4.591354.59135
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2640(1871,⋅)\chi_{2640} (1871, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2640, ( :1/2), −0.934+0.356i)(2,\ 2640,\ (\ :1/2),\ -0.934 + 0.356i)

Particular Values

L(1)L(1) ≈\approx 1.2975584021.297558402
L(12)L(\frac12) ≈\approx 1.2975584021.297558402
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+(−0.618−1.61i)T 1 + (-0.618 - 1.61i)T
5 1−iT 1 - iT
11 1+T 1 + T
good7 1−4.20iT−7T2 1 - 4.20iT - 7T^{2}
13 1−1.36T+13T2 1 - 1.36T + 13T^{2}
17 1+2.20iT−17T2 1 + 2.20iT - 17T^{2}
19 1−1.62iT−19T2 1 - 1.62iT - 19T^{2}
23 1+5.20T+23T2 1 + 5.20T + 23T^{2}
29 1−0.371iT−29T2 1 - 0.371iT - 29T^{2}
31 1−7.20iT−31T2 1 - 7.20iT - 31T^{2}
37 1+0.0213T+37T2 1 + 0.0213T + 37T^{2}
41 1+3.62iT−41T2 1 + 3.62iT - 41T^{2}
43 1−5.47iT−43T2 1 - 5.47iT - 43T^{2}
47 1−3.96T+47T2 1 - 3.96T + 47T^{2}
53 1+0.0345iT−53T2 1 + 0.0345iT - 53T^{2}
59 1−2.43T+59T2 1 - 2.43T + 59T^{2}
61 1+0.798T+61T2 1 + 0.798T + 61T^{2}
67 1+8.41iT−67T2 1 + 8.41iT - 67T^{2}
71 1−11.6T+71T2 1 - 11.6T + 71T^{2}
73 1−4.57T+73T2 1 - 4.57T + 73T^{2}
79 1−1.95iT−79T2 1 - 1.95iT - 79T^{2}
83 1+6.57T+83T2 1 + 6.57T + 83T^{2}
89 1−10.6iT−89T2 1 - 10.6iT - 89T^{2}
97 1+18.1T+97T2 1 + 18.1T + 97T^{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

−9.399749696832010123949895557494, −8.454098414017025227629159899812, −8.176110793372239986039668532070, −6.98063256772178150234811498635, −5.92945768685158629474068981846, −5.47741249186811278595295212878, −4.59453335869804399976208175469, −3.53249291214319116741784186037, −2.79206195586334672533755020431, −2.00252856434794041419028358406, 0.39821122029245242620122152421, 1.35651307412744657691515809676, 2.42769896921142298007178416703, 3.71019333077412949615137409747, 4.20059428634516804902437175527, 5.44100717620530208951935486923, 6.31109266491811245901861150589, 6.99607150973478455939218304075, 7.80857998645194361632358071123, 8.135886911258754920520668684767

Graph of the ZZ-function along the critical line