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Properties

Label 2-392-392.109-c1-0-40
Degree 22
Conductor 392392
Sign 0.207+0.978i0.207 + 0.978i
Analytic cond. 3.130133.13013
Root an. cond. 1.769211.76921
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  + (1.04 − 0.957i)2-s + (−0.167 − 1.11i)3-s + (0.165 − 1.99i)4-s + (2.65 + 1.04i)5-s + (−1.23 − 0.995i)6-s + (2.56 + 0.648i)7-s + (−1.73 − 2.23i)8-s + (1.66 − 0.512i)9-s + (3.75 − 1.45i)10-s + (−1.29 + 4.20i)11-s + (−2.24 + 0.150i)12-s + (−1.42 + 0.324i)13-s + (3.29 − 1.78i)14-s + (0.711 − 3.11i)15-s + (−3.94 − 0.658i)16-s + (−0.726 + 0.495i)17-s + ⋯
L(s)  = 1  + (0.735 − 0.677i)2-s + (−0.0966 − 0.641i)3-s + (0.0826 − 0.996i)4-s + (1.18 + 0.465i)5-s + (−0.505 − 0.406i)6-s + (0.969 + 0.245i)7-s + (−0.614 − 0.789i)8-s + (0.553 − 0.170i)9-s + (1.18 − 0.460i)10-s + (−0.390 + 1.26i)11-s + (−0.646 + 0.0433i)12-s + (−0.394 + 0.0900i)13-s + (0.879 − 0.476i)14-s + (0.183 − 0.804i)15-s + (−0.986 − 0.164i)16-s + (−0.176 + 0.120i)17-s + ⋯

Functional equation

Λ(s)=(392s/2 ΓC(s) L(s)=((0.207+0.978i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.207 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(392s/2 ΓC(s+1/2) L(s)=((0.207+0.978i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.207 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 392392    =    23⋅722^{3} \cdot 7^{2}
Sign: 0.207+0.978i0.207 + 0.978i
Analytic conductor: 3.130133.13013
Root analytic conductor: 1.769211.76921
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ392(109,⋅)\chi_{392} (109, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 392, ( :1/2), 0.207+0.978i)(2,\ 392,\ (\ :1/2),\ 0.207 + 0.978i)

Particular Values

L(1)L(1) ≈\approx 1.86302−1.50991i1.86302 - 1.50991i
L(12)L(\frac12) ≈\approx 1.86302−1.50991i1.86302 - 1.50991i
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+(−1.04+0.957i)T 1 + (-1.04 + 0.957i)T
7 1+(−2.56−0.648i)T 1 + (-2.56 - 0.648i)T
good3 1+(0.167+1.11i)T+(−2.86+0.884i)T2 1 + (0.167 + 1.11i)T + (-2.86 + 0.884i)T^{2}
5 1+(−2.65−1.04i)T+(3.66+3.40i)T2 1 + (-2.65 - 1.04i)T + (3.66 + 3.40i)T^{2}
11 1+(1.29−4.20i)T+(−9.08−6.19i)T2 1 + (1.29 - 4.20i)T + (-9.08 - 6.19i)T^{2}
13 1+(1.42−0.324i)T+(11.7−5.64i)T2 1 + (1.42 - 0.324i)T + (11.7 - 5.64i)T^{2}
17 1+(0.726−0.495i)T+(6.21−15.8i)T2 1 + (0.726 - 0.495i)T + (6.21 - 15.8i)T^{2}
19 1+(4.43−2.55i)T+(9.5−16.4i)T2 1 + (4.43 - 2.55i)T + (9.5 - 16.4i)T^{2}
23 1+(4.97+3.39i)T+(8.40+21.4i)T2 1 + (4.97 + 3.39i)T + (8.40 + 21.4i)T^{2}
29 1+(−2.61+5.42i)T+(−18.0−22.6i)T2 1 + (-2.61 + 5.42i)T + (-18.0 - 22.6i)T^{2}
31 1+(3.42−5.93i)T+(−15.5−26.8i)T2 1 + (3.42 - 5.93i)T + (-15.5 - 26.8i)T^{2}
37 1+(3.38−0.253i)T+(36.5−5.51i)T2 1 + (3.38 - 0.253i)T + (36.5 - 5.51i)T^{2}
41 1+(−2.18−2.74i)T+(−9.12+39.9i)T2 1 + (-2.18 - 2.74i)T + (-9.12 + 39.9i)T^{2}
43 1+(6.60+5.26i)T+(9.56+41.9i)T2 1 + (6.60 + 5.26i)T + (9.56 + 41.9i)T^{2}
47 1+(1.71−1.59i)T+(3.51−46.8i)T2 1 + (1.71 - 1.59i)T + (3.51 - 46.8i)T^{2}
53 1+(10.7+0.808i)T+(52.4+7.89i)T2 1 + (10.7 + 0.808i)T + (52.4 + 7.89i)T^{2}
59 1+(−10.0+3.94i)T+(43.2−40.1i)T2 1 + (-10.0 + 3.94i)T + (43.2 - 40.1i)T^{2}
61 1+(−8.05+0.603i)T+(60.3−9.09i)T2 1 + (-8.05 + 0.603i)T + (60.3 - 9.09i)T^{2}
67 1+(−11.5−6.66i)T+(33.5+58.0i)T2 1 + (-11.5 - 6.66i)T + (33.5 + 58.0i)T^{2}
71 1+(−1.46+0.703i)T+(44.2−55.5i)T2 1 + (-1.46 + 0.703i)T + (44.2 - 55.5i)T^{2}
73 1+(−11.2−10.4i)T+(5.45+72.7i)T2 1 + (-11.2 - 10.4i)T + (5.45 + 72.7i)T^{2}
79 1+(5.25+9.11i)T+(−39.5+68.4i)T2 1 + (5.25 + 9.11i)T + (-39.5 + 68.4i)T^{2}
83 1+(−3.53−0.806i)T+(74.7+36.0i)T2 1 + (-3.53 - 0.806i)T + (74.7 + 36.0i)T^{2}
89 1+(−5.35+1.65i)T+(73.5−50.1i)T2 1 + (-5.35 + 1.65i)T + (73.5 - 50.1i)T^{2}
97 1+9.24T+97T2 1 + 9.24T + 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

−11.19647822062277766343500405017, −10.05587751981127390315821902056, −9.933196904723043499606871764340, −8.320499638314972131317876793266, −6.98960982007924437417941156160, −6.27886731658997180159061124573, −5.17843682803193248857465203136, −4.21617370190446278153416152903, −2.19156901180089521942046125765, −1.87324056440732973272239883689, 2.09022796974832861510451386876, 3.78599105118565999125549991727, 4.93903306898490690990161792157, 5.42845550981580426938024729183, 6.52475803276192940514151692891, 7.78906710781726003287559279605, 8.639528333566361205855348903102, 9.611404662289265567371609535426, 10.70073825886027327554280662935, 11.42007122807183435190564014940

Graph of the ZZ-function along the critical line