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Properties

Label 2-399-19.17-c1-0-5
Degree 22
Conductor 399399
Sign −0.0745−0.997i-0.0745 - 0.997i
Analytic cond. 3.186033.18603
Root an. cond. 1.784941.78494
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  + (−0.124 + 0.705i)2-s + (−0.766 − 0.642i)3-s + (1.39 + 0.508i)4-s + (−0.780 + 0.283i)5-s + (0.548 − 0.460i)6-s + (0.5 + 0.866i)7-s + (−1.24 + 2.16i)8-s + (0.173 + 0.984i)9-s + (−0.103 − 0.585i)10-s + (−1.78 + 3.08i)11-s + (−0.743 − 1.28i)12-s + (0.608 − 0.510i)13-s + (−0.673 + 0.245i)14-s + (0.780 + 0.283i)15-s + (0.907 + 0.761i)16-s + (−0.0572 + 0.324i)17-s + ⋯
L(s)  = 1  + (−0.0879 + 0.498i)2-s + (−0.442 − 0.371i)3-s + (0.698 + 0.254i)4-s + (−0.348 + 0.126i)5-s + (0.224 − 0.187i)6-s + (0.188 + 0.327i)7-s + (−0.441 + 0.764i)8-s + (0.0578 + 0.328i)9-s + (−0.0326 − 0.185i)10-s + (−0.536 + 0.930i)11-s + (−0.214 − 0.371i)12-s + (0.168 − 0.141i)13-s + (−0.179 + 0.0654i)14-s + (0.201 + 0.0733i)15-s + (0.226 + 0.190i)16-s + (−0.0138 + 0.0787i)17-s + ⋯

Functional equation

Λ(s)=(399s/2 ΓC(s) L(s)=((−0.0745−0.997i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0745 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(399s/2 ΓC(s+1/2) L(s)=((−0.0745−0.997i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0745 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 399399    =    3⋅7⋅193 \cdot 7 \cdot 19
Sign: −0.0745−0.997i-0.0745 - 0.997i
Analytic conductor: 3.186033.18603
Root analytic conductor: 1.784941.78494
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ399(169,⋅)\chi_{399} (169, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 399, ( :1/2), −0.0745−0.997i)(2,\ 399,\ (\ :1/2),\ -0.0745 - 0.997i)

Particular Values

L(1)L(1) ≈\approx 0.805730+0.868246i0.805730 + 0.868246i
L(12)L(\frac12) ≈\approx 0.805730+0.868246i0.805730 + 0.868246i
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)
bad3 1+(0.766+0.642i)T 1 + (0.766 + 0.642i)T
7 1+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
19 1+(−3.95−1.82i)T 1 + (-3.95 - 1.82i)T
good2 1+(0.124−0.705i)T+(−1.87−0.684i)T2 1 + (0.124 - 0.705i)T + (-1.87 - 0.684i)T^{2}
5 1+(0.780−0.283i)T+(3.83−3.21i)T2 1 + (0.780 - 0.283i)T + (3.83 - 3.21i)T^{2}
11 1+(1.78−3.08i)T+(−5.5−9.52i)T2 1 + (1.78 - 3.08i)T + (-5.5 - 9.52i)T^{2}
13 1+(−0.608+0.510i)T+(2.25−12.8i)T2 1 + (-0.608 + 0.510i)T + (2.25 - 12.8i)T^{2}
17 1+(0.0572−0.324i)T+(−15.9−5.81i)T2 1 + (0.0572 - 0.324i)T + (-15.9 - 5.81i)T^{2}
23 1+(−3.33−1.21i)T+(17.6+14.7i)T2 1 + (-3.33 - 1.21i)T + (17.6 + 14.7i)T^{2}
29 1+(−0.746−4.23i)T+(−27.2+9.91i)T2 1 + (-0.746 - 4.23i)T + (-27.2 + 9.91i)T^{2}
31 1+(0.512+0.887i)T+(−15.5+26.8i)T2 1 + (0.512 + 0.887i)T + (-15.5 + 26.8i)T^{2}
37 1−0.180T+37T2 1 - 0.180T + 37T^{2}
41 1+(1.39+1.17i)T+(7.11+40.3i)T2 1 + (1.39 + 1.17i)T + (7.11 + 40.3i)T^{2}
43 1+(−2.89+1.05i)T+(32.9−27.6i)T2 1 + (-2.89 + 1.05i)T + (32.9 - 27.6i)T^{2}
47 1+(0.317+1.80i)T+(−44.1+16.0i)T2 1 + (0.317 + 1.80i)T + (-44.1 + 16.0i)T^{2}
53 1+(5.35+1.95i)T+(40.6+34.0i)T2 1 + (5.35 + 1.95i)T + (40.6 + 34.0i)T^{2}
59 1+(−1.16+6.63i)T+(−55.4−20.1i)T2 1 + (-1.16 + 6.63i)T + (-55.4 - 20.1i)T^{2}
61 1+(−11.8−4.32i)T+(46.7+39.2i)T2 1 + (-11.8 - 4.32i)T + (46.7 + 39.2i)T^{2}
67 1+(1.26+7.18i)T+(−62.9+22.9i)T2 1 + (1.26 + 7.18i)T + (-62.9 + 22.9i)T^{2}
71 1+(−12.3+4.48i)T+(54.3−45.6i)T2 1 + (-12.3 + 4.48i)T + (54.3 - 45.6i)T^{2}
73 1+(2.95+2.47i)T+(12.6+71.8i)T2 1 + (2.95 + 2.47i)T + (12.6 + 71.8i)T^{2}
79 1+(4.44+3.72i)T+(13.7+77.7i)T2 1 + (4.44 + 3.72i)T + (13.7 + 77.7i)T^{2}
83 1+(4.77+8.27i)T+(−41.5+71.8i)T2 1 + (4.77 + 8.27i)T + (-41.5 + 71.8i)T^{2}
89 1+(−0.984+0.825i)T+(15.4−87.6i)T2 1 + (-0.984 + 0.825i)T + (15.4 - 87.6i)T^{2}
97 1+(−1.42+8.09i)T+(−91.1−33.1i)T2 1 + (-1.42 + 8.09i)T + (-91.1 - 33.1i)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

−11.54063613350801273386380293111, −10.84810653699289035589796310152, −9.723819243854628117057063221395, −8.436913578142831555511016343522, −7.54587220328654743053985292726, −7.03714049229051410708213448764, −5.86603543731709809846385080341, −5.01884282925414518313468630088, −3.30095893258468043358184012989, −1.90156279304206491184500848851, 0.859373239360299371767395427317, 2.72077140799961471935681494693, 3.88583896661463370961833629629, 5.22384094051104571877475279067, 6.20790440168368072479802610701, 7.24145331148411774363013242801, 8.301607556780356469894245820368, 9.519200804262092616692894299697, 10.32815766041535388147679044964, 11.21579316658703181551763880223

Graph of the ZZ-function along the critical line