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Properties

Label 2-399-7.2-c1-0-16
Degree 22
Conductor 399399
Sign 0.100+0.994i0.100 + 0.994i
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  + (0.676 − 1.17i)2-s + (−0.5 − 0.866i)3-s + (0.0838 + 0.145i)4-s + (0.583 − 1.01i)5-s − 1.35·6-s + (1.40 + 2.24i)7-s + 2.93·8-s + (−0.499 + 0.866i)9-s + (−0.790 − 1.36i)10-s + (−3.00 − 5.20i)11-s + (0.0838 − 0.145i)12-s + 6.00·13-s + (3.57 − 0.132i)14-s − 1.16·15-s + (1.81 − 3.14i)16-s + (−1.5 − 2.59i)17-s + ⋯
L(s)  = 1  + (0.478 − 0.828i)2-s + (−0.288 − 0.499i)3-s + (0.0419 + 0.0725i)4-s + (0.261 − 0.452i)5-s − 0.552·6-s + (0.531 + 0.846i)7-s + 1.03·8-s + (−0.166 + 0.288i)9-s + (−0.249 − 0.432i)10-s + (−0.905 − 1.56i)11-s + (0.0241 − 0.0419i)12-s + 1.66·13-s + (0.956 − 0.0355i)14-s − 0.301·15-s + (0.454 − 0.787i)16-s + (−0.363 − 0.630i)17-s + ⋯

Functional equation

Λ(s)=(399s/2 ΓC(s) L(s)=((0.100+0.994i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.100 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(399s/2 ΓC(s+1/2) L(s)=((0.100+0.994i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.100 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 399399    =    3⋅7⋅193 \cdot 7 \cdot 19
Sign: 0.100+0.994i0.100 + 0.994i
Analytic conductor: 3.186033.18603
Root analytic conductor: 1.784941.78494
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ399(58,⋅)\chi_{399} (58, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 399, ( :1/2), 0.100+0.994i)(2,\ 399,\ (\ :1/2),\ 0.100 + 0.994i)

Particular Values

L(1)L(1) ≈\approx 1.41416−1.27875i1.41416 - 1.27875i
L(12)L(\frac12) ≈\approx 1.41416−1.27875i1.41416 - 1.27875i
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+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
7 1+(−1.40−2.24i)T 1 + (-1.40 - 2.24i)T
19 1+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
good2 1+(−0.676+1.17i)T+(−1−1.73i)T2 1 + (-0.676 + 1.17i)T + (-1 - 1.73i)T^{2}
5 1+(−0.583+1.01i)T+(−2.5−4.33i)T2 1 + (-0.583 + 1.01i)T + (-2.5 - 4.33i)T^{2}
11 1+(3.00+5.20i)T+(−5.5+9.52i)T2 1 + (3.00 + 5.20i)T + (-5.5 + 9.52i)T^{2}
13 1−6.00T+13T2 1 - 6.00T + 13T^{2}
17 1+(1.5+2.59i)T+(−8.5+14.7i)T2 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2}
23 1+(3.24−5.61i)T+(−11.5−19.9i)T2 1 + (3.24 - 5.61i)T + (-11.5 - 19.9i)T^{2}
29 1+6.13T+29T2 1 + 6.13T + 29T^{2}
31 1+(3.73+6.47i)T+(−15.5+26.8i)T2 1 + (3.73 + 6.47i)T + (-15.5 + 26.8i)T^{2}
37 1+(1.82−3.15i)T+(−18.5−32.0i)T2 1 + (1.82 - 3.15i)T + (-18.5 - 32.0i)T^{2}
41 1−4.77T+41T2 1 - 4.77T + 41T^{2}
43 1+0.507T+43T2 1 + 0.507T + 43T^{2}
47 1+(−2.68+4.64i)T+(−23.5−40.7i)T2 1 + (-2.68 + 4.64i)T + (-23.5 - 40.7i)T^{2}
53 1+(−3.65−6.32i)T+(−26.5+45.8i)T2 1 + (-3.65 - 6.32i)T + (-26.5 + 45.8i)T^{2}
59 1+(−4.44−7.69i)T+(−29.5+51.0i)T2 1 + (-4.44 - 7.69i)T + (-29.5 + 51.0i)T^{2}
61 1+(0.670−1.16i)T+(−30.5−52.8i)T2 1 + (0.670 - 1.16i)T + (-30.5 - 52.8i)T^{2}
67 1+(−2.18−3.78i)T+(−33.5+58.0i)T2 1 + (-2.18 - 3.78i)T + (-33.5 + 58.0i)T^{2}
71 1+9.92T+71T2 1 + 9.92T + 71T^{2}
73 1+(1.94+3.36i)T+(−36.5+63.2i)T2 1 + (1.94 + 3.36i)T + (-36.5 + 63.2i)T^{2}
79 1+(5.34−9.25i)T+(−39.5−68.4i)T2 1 + (5.34 - 9.25i)T + (-39.5 - 68.4i)T^{2}
83 1+3.07T+83T2 1 + 3.07T + 83T^{2}
89 1+(4.52−7.84i)T+(−44.5−77.0i)T2 1 + (4.52 - 7.84i)T + (-44.5 - 77.0i)T^{2}
97 1+1.01T+97T2 1 + 1.01T + 97T^{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

−11.35952779457341562436656901196, −10.70216759495511674388134809487, −9.129085213959679608489678914731, −8.343549499600599147441256650883, −7.47288209199241874365150535971, −5.83195245891497576381909975016, −5.40746633629249302237102942028, −3.83024396406843420419434859585, −2.66646155700072771625035334541, −1.37845946394378411367482400541, 1.84755085270919642383975048142, 3.93731559895141115478543231516, 4.71756908049138851120169371514, 5.77966728844986376465364917432, 6.67419952732410032522352563343, 7.48949172030391331250162572302, 8.514649475607647953949101341148, 10.04836624544270733273147257636, 10.59563955945154794593670533975, 11.07650188316144119457050417513

Graph of the ZZ-function along the critical line