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Properties

Label 2-666-9.4-c1-0-9
Degree 22
Conductor 666666
Sign −0.0351−0.999i-0.0351 - 0.999i
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  + (−0.5 − 0.866i)2-s + (−0.120 + 1.72i)3-s + (−0.499 + 0.866i)4-s + (0.181 − 0.314i)5-s + (1.55 − 0.759i)6-s + (2.51 + 4.36i)7-s + 0.999·8-s + (−2.97 − 0.416i)9-s − 0.363·10-s + (1.40 + 2.42i)11-s + (−1.43 − 0.968i)12-s + (1.98 − 3.44i)13-s + (2.51 − 4.36i)14-s + (0.521 + 0.351i)15-s + (−0.5 − 0.866i)16-s − 6.69·17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.0696 + 0.997i)3-s + (−0.249 + 0.433i)4-s + (0.0812 − 0.140i)5-s + (0.635 − 0.310i)6-s + (0.952 + 1.64i)7-s + 0.353·8-s + (−0.990 − 0.138i)9-s − 0.114·10-s + (0.422 + 0.731i)11-s + (−0.414 − 0.279i)12-s + (0.550 − 0.954i)13-s + (0.673 − 1.16i)14-s + (0.134 + 0.0908i)15-s + (−0.125 − 0.216i)16-s − 1.62·17-s + ⋯

Functional equation

Λ(s)=(666s/2 ΓC(s) L(s)=((−0.0351−0.999i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0351 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(666s/2 ΓC(s+1/2) L(s)=((−0.0351−0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0351 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 666666    =    2⋅32⋅372 \cdot 3^{2} \cdot 37
Sign: −0.0351−0.999i-0.0351 - 0.999i
Analytic conductor: 5.318035.31803
Root analytic conductor: 2.306082.30608
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ666(445,⋅)\chi_{666} (445, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 666, ( :1/2), −0.0351−0.999i)(2,\ 666,\ (\ :1/2),\ -0.0351 - 0.999i)

Particular Values

L(1)L(1) ≈\approx 0.808328+0.837296i0.808328 + 0.837296i
L(12)L(\frac12) ≈\approx 0.808328+0.837296i0.808328 + 0.837296i
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+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
3 1+(0.120−1.72i)T 1 + (0.120 - 1.72i)T
37 1+T 1 + T
good5 1+(−0.181+0.314i)T+(−2.5−4.33i)T2 1 + (-0.181 + 0.314i)T + (-2.5 - 4.33i)T^{2}
7 1+(−2.51−4.36i)T+(−3.5+6.06i)T2 1 + (-2.51 - 4.36i)T + (-3.5 + 6.06i)T^{2}
11 1+(−1.40−2.42i)T+(−5.5+9.52i)T2 1 + (-1.40 - 2.42i)T + (-5.5 + 9.52i)T^{2}
13 1+(−1.98+3.44i)T+(−6.5−11.2i)T2 1 + (-1.98 + 3.44i)T + (-6.5 - 11.2i)T^{2}
17 1+6.69T+17T2 1 + 6.69T + 17T^{2}
19 1−4.01T+19T2 1 - 4.01T + 19T^{2}
23 1+(2.61−4.53i)T+(−11.5−19.9i)T2 1 + (2.61 - 4.53i)T + (-11.5 - 19.9i)T^{2}
29 1+(3.09+5.36i)T+(−14.5+25.1i)T2 1 + (3.09 + 5.36i)T + (-14.5 + 25.1i)T^{2}
31 1+(2.38−4.12i)T+(−15.5−26.8i)T2 1 + (2.38 - 4.12i)T + (-15.5 - 26.8i)T^{2}
41 1+(−0.969+1.67i)T+(−20.5−35.5i)T2 1 + (-0.969 + 1.67i)T + (-20.5 - 35.5i)T^{2}
43 1+(0.533+0.924i)T+(−21.5+37.2i)T2 1 + (0.533 + 0.924i)T + (-21.5 + 37.2i)T^{2}
47 1+(−2.40−4.16i)T+(−23.5+40.7i)T2 1 + (-2.40 - 4.16i)T + (-23.5 + 40.7i)T^{2}
53 1−8.18T+53T2 1 - 8.18T + 53T^{2}
59 1+(3.74−6.48i)T+(−29.5−51.0i)T2 1 + (3.74 - 6.48i)T + (-29.5 - 51.0i)T^{2}
61 1+(−0.597−1.03i)T+(−30.5+52.8i)T2 1 + (-0.597 - 1.03i)T + (-30.5 + 52.8i)T^{2}
67 1+(2.35−4.08i)T+(−33.5−58.0i)T2 1 + (2.35 - 4.08i)T + (-33.5 - 58.0i)T^{2}
71 1−0.867T+71T2 1 - 0.867T + 71T^{2}
73 1+7.28T+73T2 1 + 7.28T + 73T^{2}
79 1+(−4.29−7.44i)T+(−39.5+68.4i)T2 1 + (-4.29 - 7.44i)T + (-39.5 + 68.4i)T^{2}
83 1+(5.12+8.87i)T+(−41.5+71.8i)T2 1 + (5.12 + 8.87i)T + (-41.5 + 71.8i)T^{2}
89 1−16.1T+89T2 1 - 16.1T + 89T^{2}
97 1+(7.87+13.6i)T+(−48.5+84.0i)T2 1 + (7.87 + 13.6i)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

−10.83247308134525509193031634237, −9.782985827477241361526784223506, −9.000177669269998759480488223914, −8.659546890945247594735483343210, −7.53752662430550010253085925531, −5.86323890536383235690623560178, −5.20744891745346969144047572186, −4.22539131992868549465825980338, −2.96732708966190618784736774459, −1.84706988655242206325499761994, 0.73329209591714173451359218478, 1.91720193321941790790997986872, 3.85338460690090373462779535440, 4.85722982604022905847331053230, 6.24820674677084733398294324246, 6.82944086904077452892662157402, 7.53549398852154690967042809015, 8.424070223600582542800241331233, 9.066528532878749892544227588851, 10.50906798213104942628100712883

Graph of the ZZ-function along the critical line