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Properties

Label 2-666-111.17-c1-0-7
Degree 22
Conductor 666666
Sign 0.808+0.588i0.808 + 0.588i
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 + (0.980 − 0.0857i)5-s + (1.16 + 0.981i)7-s + (−0.965 − 0.258i)8-s + (0.492 − 0.852i)10-s + (3.02 + 5.24i)11-s + (−1.58 − 0.737i)13-s + (1.47 − 0.395i)14-s + (−0.766 + 0.642i)16-s + (7.17 − 3.34i)17-s + (2.99 − 2.09i)19-s + (−0.415 − 0.892i)20-s + (6.03 + 0.527i)22-s + (−0.211 − 0.788i)23-s + ⋯
L(s)  = 1  + (0.405 − 0.579i)2-s + (−0.171 − 0.469i)4-s + (0.438 − 0.0383i)5-s + (0.442 + 0.370i)7-s + (−0.341 − 0.0915i)8-s + (0.155 − 0.269i)10-s + (0.912 + 1.58i)11-s + (−0.438 − 0.204i)13-s + (0.394 − 0.105i)14-s + (−0.191 + 0.160i)16-s + (1.73 − 0.811i)17-s + (0.686 − 0.480i)19-s + (−0.0930 − 0.199i)20-s + (1.28 + 0.112i)22-s + (−0.0440 − 0.164i)23-s + ⋯

Functional equation

Λ(s)=(666s/2 ΓC(s) L(s)=((0.808+0.588i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.808 + 0.588i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(666s/2 ΓC(s+1/2) L(s)=((0.808+0.588i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.808 + 0.588i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 666666    =    2⋅32⋅372 \cdot 3^{2} \cdot 37
Sign: 0.808+0.588i0.808 + 0.588i
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), 0.808+0.588i)(2,\ 666,\ (\ :1/2),\ 0.808 + 0.588i)

Particular Values

L(1)L(1) ≈\approx 2.04042−0.664254i2.04042 - 0.664254i
L(12)L(\frac12) ≈\approx 2.04042−0.664254i2.04042 - 0.664254i
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+(1.29+5.94i)T 1 + (1.29 + 5.94i)T
good5 1+(−0.980+0.0857i)T+(4.92−0.868i)T2 1 + (-0.980 + 0.0857i)T + (4.92 - 0.868i)T^{2}
7 1+(−1.16−0.981i)T+(1.21+6.89i)T2 1 + (-1.16 - 0.981i)T + (1.21 + 6.89i)T^{2}
11 1+(−3.02−5.24i)T+(−5.5+9.52i)T2 1 + (-3.02 - 5.24i)T + (-5.5 + 9.52i)T^{2}
13 1+(1.58+0.737i)T+(8.35+9.95i)T2 1 + (1.58 + 0.737i)T + (8.35 + 9.95i)T^{2}
17 1+(−7.17+3.34i)T+(10.9−13.0i)T2 1 + (-7.17 + 3.34i)T + (10.9 - 13.0i)T^{2}
19 1+(−2.99+2.09i)T+(6.49−17.8i)T2 1 + (-2.99 + 2.09i)T + (6.49 - 17.8i)T^{2}
23 1+(0.211+0.788i)T+(−19.9+11.5i)T2 1 + (0.211 + 0.788i)T + (-19.9 + 11.5i)T^{2}
29 1+(0.314−1.17i)T+(−25.1−14.5i)T2 1 + (0.314 - 1.17i)T + (-25.1 - 14.5i)T^{2}
31 1+(−0.312+0.312i)T−31iT2 1 + (-0.312 + 0.312i)T - 31iT^{2}
41 1+(8.84−3.21i)T+(31.4−26.3i)T2 1 + (8.84 - 3.21i)T + (31.4 - 26.3i)T^{2}
43 1+(4.57+4.57i)T+43iT2 1 + (4.57 + 4.57i)T + 43iT^{2}
47 1+(−11.5−6.65i)T+(23.5+40.7i)T2 1 + (-11.5 - 6.65i)T + (23.5 + 40.7i)T^{2}
53 1+(−7.93−9.45i)T+(−9.20+52.1i)T2 1 + (-7.93 - 9.45i)T + (-9.20 + 52.1i)T^{2}
59 1+(−0.977+11.1i)T+(−58.1−10.2i)T2 1 + (-0.977 + 11.1i)T + (-58.1 - 10.2i)T^{2}
61 1+(−3.22+6.92i)T+(−39.2−46.7i)T2 1 + (-3.22 + 6.92i)T + (-39.2 - 46.7i)T^{2}
67 1+(3.99−4.75i)T+(−11.6−65.9i)T2 1 + (3.99 - 4.75i)T + (-11.6 - 65.9i)T^{2}
71 1+(7.37+1.30i)T+(66.7+24.2i)T2 1 + (7.37 + 1.30i)T + (66.7 + 24.2i)T^{2}
73 1−8.51iT−73T2 1 - 8.51iT - 73T^{2}
79 1+(0.00716+0.0819i)T+(−77.7+13.7i)T2 1 + (0.00716 + 0.0819i)T + (-77.7 + 13.7i)T^{2}
83 1+(0.0951−0.261i)T+(−63.5−53.3i)T2 1 + (0.0951 - 0.261i)T + (-63.5 - 53.3i)T^{2}
89 1+(9.92+0.867i)T+(87.6+15.4i)T2 1 + (9.92 + 0.867i)T + (87.6 + 15.4i)T^{2}
97 1+(7.26−1.94i)T+(84.0−48.5i)T2 1 + (7.26 - 1.94i)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

−10.25156098253483317755717969806, −9.733227458943813345682160479273, −9.056582602989055659999903884529, −7.68136576834503148931329894350, −6.92214373193147978135385497727, −5.56864172832499023975730233694, −4.98848094677275224843945447663, −3.82930106878101025541759297021, −2.52530826181311822973527144469, −1.42413750434857925482767593992, 1.35372191069369371500901576974, 3.23284629402572855853503366775, 4.02938017678222074196492738033, 5.46750448486327690999220225451, 5.91633879809549021071069685426, 7.01072513874202719944276280990, 7.986448370049164324448739710564, 8.638779169439742551345137869880, 9.763525534743393404384287773540, 10.49696653840477561774229737251

Graph of the ZZ-function along the critical line