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Properties

Label 2-392-392.109-c1-0-30
Degree 22
Conductor 392392
Sign −0.0956−0.995i-0.0956 - 0.995i
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  + (1.36 + 0.354i)2-s + (0.378 + 2.50i)3-s + (1.74 + 0.971i)4-s + (1.47 + 0.580i)5-s + (−0.372 + 3.56i)6-s + (−2.45 − 0.977i)7-s + (2.04 + 1.95i)8-s + (−3.28 + 1.01i)9-s + (1.81 + 1.31i)10-s + (1.04 − 3.38i)11-s + (−1.77 + 4.75i)12-s + (−2.45 + 0.561i)13-s + (−3.01 − 2.21i)14-s + (−0.896 + 3.92i)15-s + (2.11 + 3.39i)16-s + (4.62 − 3.15i)17-s + ⋯
L(s)  = 1  + (0.967 + 0.251i)2-s + (0.218 + 1.44i)3-s + (0.873 + 0.485i)4-s + (0.661 + 0.259i)5-s + (−0.152 + 1.45i)6-s + (−0.929 − 0.369i)7-s + (0.724 + 0.689i)8-s + (−1.09 + 0.337i)9-s + (0.574 + 0.417i)10-s + (0.315 − 1.02i)11-s + (−0.512 + 1.37i)12-s + (−0.681 + 0.155i)13-s + (−0.806 − 0.590i)14-s + (−0.231 + 1.01i)15-s + (0.527 + 0.849i)16-s + (1.12 − 0.765i)17-s + ⋯

Functional equation

Λ(s)=(392s/2 ΓC(s) L(s)=((−0.0956−0.995i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0956 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(392s/2 ΓC(s+1/2) L(s)=((−0.0956−0.995i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0956 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 392392    =    23⋅722^{3} \cdot 7^{2}
Sign: −0.0956−0.995i-0.0956 - 0.995i
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), −0.0956−0.995i)(2,\ 392,\ (\ :1/2),\ -0.0956 - 0.995i)

Particular Values

L(1)L(1) ≈\approx 1.76619+1.94401i1.76619 + 1.94401i
L(12)L(\frac12) ≈\approx 1.76619+1.94401i1.76619 + 1.94401i
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+(−1.36−0.354i)T 1 + (-1.36 - 0.354i)T
7 1+(2.45+0.977i)T 1 + (2.45 + 0.977i)T
good3 1+(−0.378−2.50i)T+(−2.86+0.884i)T2 1 + (-0.378 - 2.50i)T + (-2.86 + 0.884i)T^{2}
5 1+(−1.47−0.580i)T+(3.66+3.40i)T2 1 + (-1.47 - 0.580i)T + (3.66 + 3.40i)T^{2}
11 1+(−1.04+3.38i)T+(−9.08−6.19i)T2 1 + (-1.04 + 3.38i)T + (-9.08 - 6.19i)T^{2}
13 1+(2.45−0.561i)T+(11.7−5.64i)T2 1 + (2.45 - 0.561i)T + (11.7 - 5.64i)T^{2}
17 1+(−4.62+3.15i)T+(6.21−15.8i)T2 1 + (-4.62 + 3.15i)T + (6.21 - 15.8i)T^{2}
19 1+(−1.29+0.746i)T+(9.5−16.4i)T2 1 + (-1.29 + 0.746i)T + (9.5 - 16.4i)T^{2}
23 1+(7.00+4.77i)T+(8.40+21.4i)T2 1 + (7.00 + 4.77i)T + (8.40 + 21.4i)T^{2}
29 1+(−0.882+1.83i)T+(−18.0−22.6i)T2 1 + (-0.882 + 1.83i)T + (-18.0 - 22.6i)T^{2}
31 1+(4.73−8.20i)T+(−15.5−26.8i)T2 1 + (4.73 - 8.20i)T + (-15.5 - 26.8i)T^{2}
37 1+(−6.56+0.492i)T+(36.5−5.51i)T2 1 + (-6.56 + 0.492i)T + (36.5 - 5.51i)T^{2}
41 1+(−3.66−4.59i)T+(−9.12+39.9i)T2 1 + (-3.66 - 4.59i)T + (-9.12 + 39.9i)T^{2}
43 1+(0.520+0.415i)T+(9.56+41.9i)T2 1 + (0.520 + 0.415i)T + (9.56 + 41.9i)T^{2}
47 1+(−1.14+1.06i)T+(3.51−46.8i)T2 1 + (-1.14 + 1.06i)T + (3.51 - 46.8i)T^{2}
53 1+(5.33+0.399i)T+(52.4+7.89i)T2 1 + (5.33 + 0.399i)T + (52.4 + 7.89i)T^{2}
59 1+(4.93−1.93i)T+(43.2−40.1i)T2 1 + (4.93 - 1.93i)T + (43.2 - 40.1i)T^{2}
61 1+(−11.6+0.875i)T+(60.3−9.09i)T2 1 + (-11.6 + 0.875i)T + (60.3 - 9.09i)T^{2}
67 1+(0.113+0.0657i)T+(33.5+58.0i)T2 1 + (0.113 + 0.0657i)T + (33.5 + 58.0i)T^{2}
71 1+(2.61−1.26i)T+(44.2−55.5i)T2 1 + (2.61 - 1.26i)T + (44.2 - 55.5i)T^{2}
73 1+(11.9+11.0i)T+(5.45+72.7i)T2 1 + (11.9 + 11.0i)T + (5.45 + 72.7i)T^{2}
79 1+(6.19+10.7i)T+(−39.5+68.4i)T2 1 + (6.19 + 10.7i)T + (-39.5 + 68.4i)T^{2}
83 1+(−5.86−1.33i)T+(74.7+36.0i)T2 1 + (-5.86 - 1.33i)T + (74.7 + 36.0i)T^{2}
89 1+(−10.3+3.19i)T+(73.5−50.1i)T2 1 + (-10.3 + 3.19i)T + (73.5 - 50.1i)T^{2}
97 1+10.3T+97T2 1 + 10.3T + 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

−11.54747188678210113281845631010, −10.42584875565952155398164523471, −9.981978270828879551484444061240, −9.041809171339824901385289809423, −7.70417884858977742619758014992, −6.43598661946149498457777375387, −5.67479984011243041145232871963, −4.57487007604418968533393755476, −3.56383218078167661985917469317, −2.77244135227625530937723396968, 1.58838614106480375459109232487, 2.46127558034480351842700048778, 3.87284117407338096427752000483, 5.60764222981453650439186361076, 6.07976840042043219676213381251, 7.20066996275840347935832597247, 7.81633306151332635530600799780, 9.590649540978766784279971074344, 9.967856195881801075828326636790, 11.61129991927956220906422817189

Graph of the ZZ-function along the critical line