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Properties

Label 2-666-111.17-c1-0-11
Degree 22
Conductor 666666
Sign −0.645+0.763i-0.645 + 0.763i
Analytic cond. 5.318035.31803
Root an. cond. 2.306082.30608
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  + (−0.573 + 0.819i)2-s + (−0.342 − 0.939i)4-s + (1.31 − 0.114i)5-s + (−1.79 − 1.51i)7-s + (0.965 + 0.258i)8-s + (−0.658 + 1.14i)10-s + (−1.26 − 2.19i)11-s + (−5.68 − 2.65i)13-s + (2.26 − 0.608i)14-s + (−0.766 + 0.642i)16-s + (−2.17 + 1.01i)17-s + (−5.10 + 3.57i)19-s + (−0.556 − 1.19i)20-s + (2.52 + 0.221i)22-s + (1.95 + 7.30i)23-s + ⋯
L(s)  = 1  + (−0.405 + 0.579i)2-s + (−0.171 − 0.469i)4-s + (0.587 − 0.0513i)5-s + (−0.680 − 0.570i)7-s + (0.341 + 0.0915i)8-s + (−0.208 + 0.360i)10-s + (−0.382 − 0.662i)11-s + (−1.57 − 0.735i)13-s + (0.606 − 0.162i)14-s + (−0.191 + 0.160i)16-s + (−0.526 + 0.245i)17-s + (−1.17 + 0.820i)19-s + (−0.124 − 0.267i)20-s + (0.538 + 0.0471i)22-s + (0.408 + 1.52i)23-s + ⋯

Functional equation

Λ(s)=(666s/2 ΓC(s) L(s)=((−0.645+0.763i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.645 + 0.763i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(666s/2 ΓC(s+1/2) L(s)=((−0.645+0.763i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.645 + 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 666666    =    2⋅32⋅372 \cdot 3^{2} \cdot 37
Sign: −0.645+0.763i-0.645 + 0.763i
Analytic conductor: 5.318035.31803
Root analytic conductor: 2.306082.30608
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ666(17,⋅)\chi_{666} (17, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 666, ( :1/2), −0.645+0.763i)(2,\ 666,\ (\ :1/2),\ -0.645 + 0.763i)

Particular Values

L(1)L(1) ≈\approx 0.124564−0.268276i0.124564 - 0.268276i
L(12)L(\frac12) ≈\approx 0.124564−0.268276i0.124564 - 0.268276i
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+(0.573−0.819i)T 1 + (0.573 - 0.819i)T
3 1 1
37 1+(6.02−0.866i)T 1 + (6.02 - 0.866i)T
good5 1+(−1.31+0.114i)T+(4.92−0.868i)T2 1 + (-1.31 + 0.114i)T + (4.92 - 0.868i)T^{2}
7 1+(1.79+1.51i)T+(1.21+6.89i)T2 1 + (1.79 + 1.51i)T + (1.21 + 6.89i)T^{2}
11 1+(1.26+2.19i)T+(−5.5+9.52i)T2 1 + (1.26 + 2.19i)T + (-5.5 + 9.52i)T^{2}
13 1+(5.68+2.65i)T+(8.35+9.95i)T2 1 + (5.68 + 2.65i)T + (8.35 + 9.95i)T^{2}
17 1+(2.17−1.01i)T+(10.9−13.0i)T2 1 + (2.17 - 1.01i)T + (10.9 - 13.0i)T^{2}
19 1+(5.10−3.57i)T+(6.49−17.8i)T2 1 + (5.10 - 3.57i)T + (6.49 - 17.8i)T^{2}
23 1+(−1.95−7.30i)T+(−19.9+11.5i)T2 1 + (-1.95 - 7.30i)T + (-19.9 + 11.5i)T^{2}
29 1+(−2.27+8.48i)T+(−25.1−14.5i)T2 1 + (-2.27 + 8.48i)T + (-25.1 - 14.5i)T^{2}
31 1+(−1.77+1.77i)T−31iT2 1 + (-1.77 + 1.77i)T - 31iT^{2}
41 1+(10.3−3.74i)T+(31.4−26.3i)T2 1 + (10.3 - 3.74i)T + (31.4 - 26.3i)T^{2}
43 1+(0.789+0.789i)T+43iT2 1 + (0.789 + 0.789i)T + 43iT^{2}
47 1+(−9.67−5.58i)T+(23.5+40.7i)T2 1 + (-9.67 - 5.58i)T + (23.5 + 40.7i)T^{2}
53 1+(1.18+1.41i)T+(−9.20+52.1i)T2 1 + (1.18 + 1.41i)T + (-9.20 + 52.1i)T^{2}
59 1+(−0.761+8.70i)T+(−58.1−10.2i)T2 1 + (-0.761 + 8.70i)T + (-58.1 - 10.2i)T^{2}
61 1+(−4.69+10.0i)T+(−39.2−46.7i)T2 1 + (-4.69 + 10.0i)T + (-39.2 - 46.7i)T^{2}
67 1+(−5.02+5.99i)T+(−11.6−65.9i)T2 1 + (-5.02 + 5.99i)T + (-11.6 - 65.9i)T^{2}
71 1+(4.03+0.711i)T+(66.7+24.2i)T2 1 + (4.03 + 0.711i)T + (66.7 + 24.2i)T^{2}
73 1+10.8iT−73T2 1 + 10.8iT - 73T^{2}
79 1+(−0.818−9.35i)T+(−77.7+13.7i)T2 1 + (-0.818 - 9.35i)T + (-77.7 + 13.7i)T^{2}
83 1+(−0.388+1.06i)T+(−63.5−53.3i)T2 1 + (-0.388 + 1.06i)T + (-63.5 - 53.3i)T^{2}
89 1+(−0.317−0.0277i)T+(87.6+15.4i)T2 1 + (-0.317 - 0.0277i)T + (87.6 + 15.4i)T^{2}
97 1+(−12.8+3.43i)T+(84.0−48.5i)T2 1 + (-12.8 + 3.43i)T + (84.0 - 48.5i)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

−9.973845834616758908182545601084, −9.562329293869585571755482135741, −8.314274491817895999694188738325, −7.64368442824694157496259695484, −6.62391085058109591956051028421, −5.85313489028721498177464327951, −4.92067928385727593669315550712, −3.55259856942433448798839150086, −2.11723774832707052984414094631, −0.16288020257869580931322319848, 2.15521313097231060019912821253, 2.69635650782155329414000392666, 4.37779694985330356764197615045, 5.21892978423689253660983587730, 6.69676181866716138015028267886, 7.11277555971713332823859402691, 8.729876948629715414314125475970, 9.038172596833064552719142793423, 10.18415232992137739439415636357, 10.41996433493717024790763360078

Graph of the ZZ-function along the critical line