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Properties

Label 2-1338-223.183-c1-0-2
Degree 22
Conductor 13381338
Sign −0.997−0.0712i-0.997 - 0.0712i
Analytic cond. 10.683910.6839
Root an. cond. 3.268633.26863
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  − 2-s + (−0.5 − 0.866i)3-s + 4-s + (−1.93 + 3.35i)5-s + (0.5 + 0.866i)6-s − 0.151·7-s − 8-s + (−0.499 + 0.866i)9-s + (1.93 − 3.35i)10-s + (0.443 − 0.768i)11-s + (−0.5 − 0.866i)12-s − 2.55·13-s + 0.151·14-s + 3.87·15-s + 16-s + 6.14·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.288 − 0.499i)3-s + 0.5·4-s + (−0.867 + 1.50i)5-s + (0.204 + 0.353i)6-s − 0.0574·7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (0.613 − 1.06i)10-s + (0.133 − 0.231i)11-s + (−0.144 − 0.249i)12-s − 0.709·13-s + 0.0406·14-s + 1.00·15-s + 0.250·16-s + 1.49·17-s + ⋯

Functional equation

Λ(s)=(1338s/2 ΓC(s) L(s)=((−0.997−0.0712i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0712i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1338s/2 ΓC(s+1/2) L(s)=((−0.997−0.0712i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1338 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0712i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13381338    =    2⋅3⋅2232 \cdot 3 \cdot 223
Sign: −0.997−0.0712i-0.997 - 0.0712i
Analytic conductor: 10.683910.6839
Root analytic conductor: 3.268633.26863
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1338(1075,⋅)\chi_{1338} (1075, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1338, ( :1/2), −0.997−0.0712i)(2,\ 1338,\ (\ :1/2),\ -0.997 - 0.0712i)

Particular Values

L(1)L(1) ≈\approx 0.25658432880.2565843288
L(12)L(\frac12) ≈\approx 0.25658432880.2565843288
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+T 1 + T
3 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
223 1+(−4.89−14.1i)T 1 + (-4.89 - 14.1i)T
good5 1+(1.93−3.35i)T+(−2.5−4.33i)T2 1 + (1.93 - 3.35i)T + (-2.5 - 4.33i)T^{2}
7 1+0.151T+7T2 1 + 0.151T + 7T^{2}
11 1+(−0.443+0.768i)T+(−5.5−9.52i)T2 1 + (-0.443 + 0.768i)T + (-5.5 - 9.52i)T^{2}
13 1+2.55T+13T2 1 + 2.55T + 13T^{2}
17 1−6.14T+17T2 1 - 6.14T + 17T^{2}
19 1+(−1.72−2.98i)T+(−9.5+16.4i)T2 1 + (-1.72 - 2.98i)T + (-9.5 + 16.4i)T^{2}
23 1+(−2.21−3.83i)T+(−11.5+19.9i)T2 1 + (-2.21 - 3.83i)T + (-11.5 + 19.9i)T^{2}
29 1+(3.88−6.73i)T+(−14.5−25.1i)T2 1 + (3.88 - 6.73i)T + (-14.5 - 25.1i)T^{2}
31 1+(0.208+0.360i)T+(−15.5+26.8i)T2 1 + (0.208 + 0.360i)T + (-15.5 + 26.8i)T^{2}
37 1+(−1.08+1.88i)T+(−18.5−32.0i)T2 1 + (-1.08 + 1.88i)T + (-18.5 - 32.0i)T^{2}
41 1+2.19T+41T2 1 + 2.19T + 41T^{2}
43 1+(2.79+4.83i)T+(−21.5+37.2i)T2 1 + (2.79 + 4.83i)T + (-21.5 + 37.2i)T^{2}
47 1+(5.30−9.19i)T+(−23.5−40.7i)T2 1 + (5.30 - 9.19i)T + (-23.5 - 40.7i)T^{2}
53 1+(−2.48+4.30i)T+(−26.5−45.8i)T2 1 + (-2.48 + 4.30i)T + (-26.5 - 45.8i)T^{2}
59 1+10.6T+59T2 1 + 10.6T + 59T^{2}
61 1+(7.54+13.0i)T+(−30.5+52.8i)T2 1 + (7.54 + 13.0i)T + (-30.5 + 52.8i)T^{2}
67 1+(4.36+7.56i)T+(−33.5+58.0i)T2 1 + (4.36 + 7.56i)T + (-33.5 + 58.0i)T^{2}
71 1+(−2.92−5.06i)T+(−35.5+61.4i)T2 1 + (-2.92 - 5.06i)T + (-35.5 + 61.4i)T^{2}
73 1+(−0.302+0.524i)T+(−36.5−63.2i)T2 1 + (-0.302 + 0.524i)T + (-36.5 - 63.2i)T^{2}
79 1+(−2.25+3.89i)T+(−39.5−68.4i)T2 1 + (-2.25 + 3.89i)T + (-39.5 - 68.4i)T^{2}
83 1+(2.87−4.97i)T+(−41.5−71.8i)T2 1 + (2.87 - 4.97i)T + (-41.5 - 71.8i)T^{2}
89 1+(4.67+8.09i)T+(−44.5+77.0i)T2 1 + (4.67 + 8.09i)T + (-44.5 + 77.0i)T^{2}
97 1+(−1.11+1.92i)T+(−48.5−84.0i)T2 1 + (-1.11 + 1.92i)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

−10.07186346543878077273814678525, −9.333431039019519157246635674183, −7.968537560317635557052710314646, −7.65393642673750409161538301998, −6.97937909103164617424991882735, −6.19792852822337220311704126178, −5.19657551537875337951403271980, −3.51297896950889320226041984786, −3.04134754300483396894562674460, −1.58266930369195455381521603031, 0.15261201402289831726445057485, 1.30067724739835968969549871740, 3.02105119021876405426658328150, 4.20432884488372548186933815072, 4.93782811982024684020912310196, 5.72312921052478905336570476491, 7.00974914784021826253504371658, 7.86064398410774444142960109424, 8.407576758920523784619898709078, 9.382696766655458734959943416632

Graph of the ZZ-function along the critical line