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Properties

Label 2-1323-9.7-c1-0-14
Degree 22
Conductor 13231323
Sign 0.989−0.145i0.989 - 0.145i
Analytic cond. 10.564210.5642
Root an. cond. 3.250263.25026
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  + (0.849 − 1.47i)2-s + (−0.444 − 0.769i)4-s + (1.79 + 3.10i)5-s + 1.88·8-s + 6.09·10-s + (−1.40 + 2.43i)11-s + (0.5 + 0.866i)13-s + (2.49 − 4.31i)16-s − 4.11·17-s + 0.888·19-s + (1.59 − 2.76i)20-s + (2.38 + 4.13i)22-s + (2.93 + 5.08i)23-s + (−3.93 + 6.82i)25-s + 1.69·26-s + ⋯
L(s)  = 1  + (0.600 − 1.04i)2-s + (−0.222 − 0.384i)4-s + (0.802 + 1.38i)5-s + 0.667·8-s + 1.92·10-s + (−0.423 + 0.733i)11-s + (0.138 + 0.240i)13-s + (0.623 − 1.07i)16-s − 0.997·17-s + 0.203·19-s + (0.356 − 0.617i)20-s + (0.509 + 0.882i)22-s + (0.612 + 1.06i)23-s + (−0.787 + 1.36i)25-s + 0.333·26-s + ⋯

Functional equation

Λ(s)=(1323s/2 ΓC(s) L(s)=((0.989−0.145i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.145i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1323s/2 ΓC(s+1/2) L(s)=((0.989−0.145i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 - 0.145i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13231323    =    33⋅723^{3} \cdot 7^{2}
Sign: 0.989−0.145i0.989 - 0.145i
Analytic conductor: 10.564210.5642
Root analytic conductor: 3.250263.25026
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1323(883,⋅)\chi_{1323} (883, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1323, ( :1/2), 0.989−0.145i)(2,\ 1323,\ (\ :1/2),\ 0.989 - 0.145i)

Particular Values

L(1)L(1) ≈\approx 2.7339150352.733915035
L(12)L(\frac12) ≈\approx 2.7339150352.733915035
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 1
7 1 1
good2 1+(−0.849+1.47i)T+(−1−1.73i)T2 1 + (-0.849 + 1.47i)T + (-1 - 1.73i)T^{2}
5 1+(−1.79−3.10i)T+(−2.5+4.33i)T2 1 + (-1.79 - 3.10i)T + (-2.5 + 4.33i)T^{2}
11 1+(1.40−2.43i)T+(−5.5−9.52i)T2 1 + (1.40 - 2.43i)T + (-5.5 - 9.52i)T^{2}
13 1+(−0.5−0.866i)T+(−6.5+11.2i)T2 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2}
17 1+4.11T+17T2 1 + 4.11T + 17T^{2}
19 1−0.888T+19T2 1 - 0.888T + 19T^{2}
23 1+(−2.93−5.08i)T+(−11.5+19.9i)T2 1 + (-2.93 - 5.08i)T + (-11.5 + 19.9i)T^{2}
29 1+(0.849−1.47i)T+(−14.5−25.1i)T2 1 + (0.849 - 1.47i)T + (-14.5 - 25.1i)T^{2}
31 1+(3.49+6.05i)T+(−15.5+26.8i)T2 1 + (3.49 + 6.05i)T + (-15.5 + 26.8i)T^{2}
37 1−4.76T+37T2 1 - 4.76T + 37T^{2}
41 1+(−2.70−4.68i)T+(−20.5+35.5i)T2 1 + (-2.70 - 4.68i)T + (-20.5 + 35.5i)T^{2}
43 1+(2.60−4.51i)T+(−21.5−37.2i)T2 1 + (2.60 - 4.51i)T + (-21.5 - 37.2i)T^{2}
47 1+(−1.33+2.30i)T+(−23.5−40.7i)T2 1 + (-1.33 + 2.30i)T + (-23.5 - 40.7i)T^{2}
53 1−0.123T+53T2 1 - 0.123T + 53T^{2}
59 1+(−4.43−7.68i)T+(−29.5+51.0i)T2 1 + (-4.43 - 7.68i)T + (-29.5 + 51.0i)T^{2}
61 1+(−1.93+3.35i)T+(−30.5−52.8i)T2 1 + (-1.93 + 3.35i)T + (-30.5 - 52.8i)T^{2}
67 1+(6.15+10.6i)T+(−33.5+58.0i)T2 1 + (6.15 + 10.6i)T + (-33.5 + 58.0i)T^{2}
71 1−2.87T+71T2 1 - 2.87T + 71T^{2}
73 1−10.6T+73T2 1 - 10.6T + 73T^{2}
79 1+(−3.54+6.13i)T+(−39.5−68.4i)T2 1 + (-3.54 + 6.13i)T + (-39.5 - 68.4i)T^{2}
83 1+(−2.05+3.56i)T+(−41.5−71.8i)T2 1 + (-2.05 + 3.56i)T + (-41.5 - 71.8i)T^{2}
89 1−9.60T+89T2 1 - 9.60T + 89T^{2}
97 1+(−3.66+6.34i)T+(−48.5−84.0i)T2 1 + (-3.66 + 6.34i)T + (-48.5 - 84.0i)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

−9.823481960215475884213411397038, −9.336345103023807331900836891114, −7.79568996140009908928008632609, −7.14782728326546765930230335525, −6.32783263024313804384296774459, −5.30064130726495520714260941970, −4.29560433734837518888377513956, −3.28671302026344947800047341609, −2.49022990805779582779462559220, −1.76617313827421943088596525522, 0.959147395340411066769004114683, 2.30016943899736280641930683705, 3.94056782090519340924507797513, 4.93638861308477700226044222074, 5.35631106018319872019443444613, 6.16054487269092100459347499199, 6.92030698961121385745287050834, 8.035844792324329767183263503597, 8.658859188276740665480286185771, 9.317607950618066358470961629619

Graph of the ZZ-function along the critical line