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Properties

Label 2-39-39.11-c1-0-1
Degree 22
Conductor 3939
Sign 0.351+0.936i0.351 + 0.936i
Analytic cond. 0.3114160.311416
Root an. cond. 0.5580470.558047
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  + (−1.45 − 0.389i)2-s + (0.239 − 1.71i)3-s + (0.232 + 0.133i)4-s + (1.06 − 1.06i)5-s + (−1.01 + 2.40i)6-s + (0.366 + 1.36i)7-s + (1.84 + 1.84i)8-s + (−2.88 − 0.820i)9-s + (−1.96 + 1.13i)10-s + (−1.06 + 3.97i)11-s + (0.285 − 0.366i)12-s + (3.59 + 0.232i)13-s − 2.12i·14-s + (−1.57 − 2.08i)15-s + (−2.23 − 3.86i)16-s + (2.51 − 4.36i)17-s + ⋯
L(s)  = 1  + (−1.02 − 0.275i)2-s + (0.138 − 0.990i)3-s + (0.116 + 0.0669i)4-s + (0.476 − 0.476i)5-s + (−0.415 + 0.980i)6-s + (0.138 + 0.516i)7-s + (0.652 + 0.652i)8-s + (−0.961 − 0.273i)9-s + (−0.621 + 0.358i)10-s + (−0.321 + 1.19i)11-s + (0.0823 − 0.105i)12-s + (0.997 + 0.0643i)13-s − 0.569i·14-s + (−0.405 − 0.537i)15-s + (−0.558 − 0.966i)16-s + (0.611 − 1.05i)17-s + ⋯

Functional equation

Λ(s)=(39s/2 ΓC(s) L(s)=((0.351+0.936i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.351 + 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(39s/2 ΓC(s+1/2) L(s)=((0.351+0.936i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.351 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 3939    =    3⋅133 \cdot 13
Sign: 0.351+0.936i0.351 + 0.936i
Analytic conductor: 0.3114160.311416
Root analytic conductor: 0.5580470.558047
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ39(11,⋅)\chi_{39} (11, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 39, ( :1/2), 0.351+0.936i)(2,\ 39,\ (\ :1/2),\ 0.351 + 0.936i)

Particular Values

L(1)L(1) ≈\approx 0.421043−0.291630i0.421043 - 0.291630i
L(12)L(\frac12) ≈\approx 0.421043−0.291630i0.421043 - 0.291630i
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+(−0.239+1.71i)T 1 + (-0.239 + 1.71i)T
13 1+(−3.59−0.232i)T 1 + (-3.59 - 0.232i)T
good2 1+(1.45+0.389i)T+(1.73+i)T2 1 + (1.45 + 0.389i)T + (1.73 + i)T^{2}
5 1+(−1.06+1.06i)T−5iT2 1 + (-1.06 + 1.06i)T - 5iT^{2}
7 1+(−0.366−1.36i)T+(−6.06+3.5i)T2 1 + (-0.366 - 1.36i)T + (-6.06 + 3.5i)T^{2}
11 1+(1.06−3.97i)T+(−9.52−5.5i)T2 1 + (1.06 - 3.97i)T + (-9.52 - 5.5i)T^{2}
17 1+(−2.51+4.36i)T+(−8.5−14.7i)T2 1 + (-2.51 + 4.36i)T + (-8.5 - 14.7i)T^{2}
19 1+(3.73−i)T+(16.4−9.5i)T2 1 + (3.73 - i)T + (16.4 - 9.5i)T^{2}
23 1+(−11.5+19.9i)T2 1 + (-11.5 + 19.9i)T^{2}
29 1+(6.20−3.58i)T+(14.5−25.1i)T2 1 + (6.20 - 3.58i)T + (14.5 - 25.1i)T^{2}
31 1+(2.46+2.46i)T+31iT2 1 + (2.46 + 2.46i)T + 31iT^{2}
37 1+(5.23+1.40i)T+(32.0+18.5i)T2 1 + (5.23 + 1.40i)T + (32.0 + 18.5i)T^{2}
41 1+(−5.42−1.45i)T+(35.5+20.5i)T2 1 + (-5.42 - 1.45i)T + (35.5 + 20.5i)T^{2}
43 1+(−1.90−1.09i)T+(21.5+37.2i)T2 1 + (-1.90 - 1.09i)T + (21.5 + 37.2i)T^{2}
47 1+(4.25+4.25i)T+47iT2 1 + (4.25 + 4.25i)T + 47iT^{2}
53 1+0.779iT−53T2 1 + 0.779iT - 53T^{2}
59 1+(2.90−0.779i)T+(51.0−29.5i)T2 1 + (2.90 - 0.779i)T + (51.0 - 29.5i)T^{2}
61 1+(−3.5+6.06i)T+(−30.5−52.8i)T2 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2}
67 1+(1.53−5.73i)T+(−58.0−33.5i)T2 1 + (1.53 - 5.73i)T + (-58.0 - 33.5i)T^{2}
71 1+(0.779+2.90i)T+(−61.4+35.5i)T2 1 + (0.779 + 2.90i)T + (-61.4 + 35.5i)T^{2}
73 1+(0.901−0.901i)T−73iT2 1 + (0.901 - 0.901i)T - 73iT^{2}
79 1−2T+79T2 1 - 2T + 79T^{2}
83 1+(−2.90+2.90i)T−83iT2 1 + (-2.90 + 2.90i)T - 83iT^{2}
89 1+(2.41−9.01i)T+(−77.0−44.5i)T2 1 + (2.41 - 9.01i)T + (-77.0 - 44.5i)T^{2}
97 1+(−1.63+0.437i)T+(84.0−48.5i)T2 1 + (-1.63 + 0.437i)T + (84.0 - 48.5i)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

−16.55743357445146294226353549882, −14.78615635204347166425771417192, −13.53082564970482701980803769968, −12.50570744403579383388635188000, −11.13681544742550342333398363301, −9.560487094476591237969049109584, −8.642274846801149240759873676364, −7.37041387682648272866778873398, −5.43884288613253686495927374509, −1.84816786409955800015425415144, 3.81856285014947763266534833046, 6.05142661502583991481500636523, 8.045450346075109607081572456416, 8.997619564720382669656620647576, 10.39841282818986344036276589992, 10.87756384267549753742568170851, 13.28951949705771400240830153780, 14.32023371589416464065032437022, 15.72051592257398388121122366497, 16.66238193703605810760542091106

Graph of the ZZ-function along the critical line