↑

Properties

Label 2-666-111.17-c1-0-4
Degree 22
Conductor 666666
Sign 0.903+0.429i0.903 + 0.429i
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.573 + 0.819i)2-s + (−0.342 − 0.939i)4-s + (−3.25 + 0.284i)5-s + (−0.761 − 0.638i)7-s + (0.965 + 0.258i)8-s + (1.63 − 2.83i)10-s + (2.46 + 4.26i)11-s + (−3.00 − 1.40i)13-s + (0.960 − 0.257i)14-s + (−0.766 + 0.642i)16-s + (−0.884 + 0.412i)17-s + (6.51 − 4.56i)19-s + (1.38 + 2.96i)20-s + (−4.91 − 0.429i)22-s + (−1.73 − 6.46i)23-s + ⋯
L(s)  = 1  + (−0.405 + 0.579i)2-s + (−0.171 − 0.469i)4-s + (−1.45 + 0.127i)5-s + (−0.287 − 0.241i)7-s + (0.341 + 0.0915i)8-s + (0.516 − 0.895i)10-s + (0.743 + 1.28i)11-s + (−0.833 − 0.388i)13-s + (0.256 − 0.0687i)14-s + (−0.191 + 0.160i)16-s + (−0.214 + 0.100i)17-s + (1.49 − 1.04i)19-s + (0.308 + 0.662i)20-s + (−1.04 − 0.0916i)22-s + (−0.361 − 1.34i)23-s + ⋯

Functional equation

Λ(s)=(666s/2 ΓC(s) L(s)=((0.903+0.429i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.903 + 0.429i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(666s/2 ΓC(s+1/2) L(s)=((0.903+0.429i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.903 + 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 666666    =    2⋅32⋅372 \cdot 3^{2} \cdot 37
Sign: 0.903+0.429i0.903 + 0.429i
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), 0.903+0.429i)(2,\ 666,\ (\ :1/2),\ 0.903 + 0.429i)

Particular Values

L(1)L(1) ≈\approx 0.698686−0.157674i0.698686 - 0.157674i
L(12)L(\frac12) ≈\approx 0.698686−0.157674i0.698686 - 0.157674i
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+(−2.37−5.60i)T 1 + (-2.37 - 5.60i)T
good5 1+(3.25−0.284i)T+(4.92−0.868i)T2 1 + (3.25 - 0.284i)T + (4.92 - 0.868i)T^{2}
7 1+(0.761+0.638i)T+(1.21+6.89i)T2 1 + (0.761 + 0.638i)T + (1.21 + 6.89i)T^{2}
11 1+(−2.46−4.26i)T+(−5.5+9.52i)T2 1 + (-2.46 - 4.26i)T + (-5.5 + 9.52i)T^{2}
13 1+(3.00+1.40i)T+(8.35+9.95i)T2 1 + (3.00 + 1.40i)T + (8.35 + 9.95i)T^{2}
17 1+(0.884−0.412i)T+(10.9−13.0i)T2 1 + (0.884 - 0.412i)T + (10.9 - 13.0i)T^{2}
19 1+(−6.51+4.56i)T+(6.49−17.8i)T2 1 + (-6.51 + 4.56i)T + (6.49 - 17.8i)T^{2}
23 1+(1.73+6.46i)T+(−19.9+11.5i)T2 1 + (1.73 + 6.46i)T + (-19.9 + 11.5i)T^{2}
29 1+(−1.24+4.64i)T+(−25.1−14.5i)T2 1 + (-1.24 + 4.64i)T + (-25.1 - 14.5i)T^{2}
31 1+(−7.57+7.57i)T−31iT2 1 + (-7.57 + 7.57i)T - 31iT^{2}
41 1+(−7.43+2.70i)T+(31.4−26.3i)T2 1 + (-7.43 + 2.70i)T + (31.4 - 26.3i)T^{2}
43 1+(1.60+1.60i)T+43iT2 1 + (1.60 + 1.60i)T + 43iT^{2}
47 1+(−3.88−2.24i)T+(23.5+40.7i)T2 1 + (-3.88 - 2.24i)T + (23.5 + 40.7i)T^{2}
53 1+(7.06+8.42i)T+(−9.20+52.1i)T2 1 + (7.06 + 8.42i)T + (-9.20 + 52.1i)T^{2}
59 1+(0.720−8.23i)T+(−58.1−10.2i)T2 1 + (0.720 - 8.23i)T + (-58.1 - 10.2i)T^{2}
61 1+(3.71−7.97i)T+(−39.2−46.7i)T2 1 + (3.71 - 7.97i)T + (-39.2 - 46.7i)T^{2}
67 1+(1.59−1.89i)T+(−11.6−65.9i)T2 1 + (1.59 - 1.89i)T + (-11.6 - 65.9i)T^{2}
71 1+(5.07+0.894i)T+(66.7+24.2i)T2 1 + (5.07 + 0.894i)T + (66.7 + 24.2i)T^{2}
73 1−5.03iT−73T2 1 - 5.03iT - 73T^{2}
79 1+(−0.278−3.18i)T+(−77.7+13.7i)T2 1 + (-0.278 - 3.18i)T + (-77.7 + 13.7i)T^{2}
83 1+(−3.09+8.50i)T+(−63.5−53.3i)T2 1 + (-3.09 + 8.50i)T + (-63.5 - 53.3i)T^{2}
89 1+(5.91+0.517i)T+(87.6+15.4i)T2 1 + (5.91 + 0.517i)T + (87.6 + 15.4i)T^{2}
97 1+(−11.4+3.07i)T+(84.0−48.5i)T2 1 + (-11.4 + 3.07i)T + (84.0 - 48.5i)T^{2}
show more
show less
   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

−10.16955595983152807213696210558, −9.650020448223171381491981301246, −8.567135235959343606996136074984, −7.59490730177684338060315931985, −7.23714947677144242372894337093, −6.28117842072093014980664319632, −4.71611260065913843435355536511, −4.20842937148244174874986818922, −2.69629893547749514731847645285, −0.54712928448343228198583327443, 1.12237424468986961176868979566, 3.10136977690793833510052680107, 3.68866242849198404595988976045, 4.85429871258052910201591878328, 6.17801494542483143369937597266, 7.44375960635166222944732063019, 7.951912875261990053582597622968, 8.971504096264490286735924156494, 9.577155114273327096188387629573, 10.75488841049895515156986826504

Graph of the ZZ-function along the critical line