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Properties

Label 2-1323-9.4-c1-0-32
Degree 22
Conductor 13231323
Sign −0.841+0.539i-0.841 + 0.539i
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  + (−0.247 − 0.429i)2-s + (0.877 − 1.51i)4-s + (1.84 − 3.19i)5-s − 1.86·8-s − 1.83·10-s + (−0.446 − 0.772i)11-s + (−0.598 + 1.03i)13-s + (−1.29 − 2.23i)16-s + 0.249·17-s + 2.80·19-s + (−3.23 − 5.60i)20-s + (−0.221 + 0.383i)22-s + (1.23 − 2.14i)23-s + (−4.31 − 7.47i)25-s + 0.593·26-s + ⋯
L(s)  = 1  + (−0.175 − 0.303i)2-s + (0.438 − 0.759i)4-s + (0.825 − 1.43i)5-s − 0.658·8-s − 0.579·10-s + (−0.134 − 0.233i)11-s + (−0.165 + 0.287i)13-s + (−0.323 − 0.559i)16-s + 0.0606·17-s + 0.644·19-s + (−0.724 − 1.25i)20-s + (−0.0471 + 0.0817i)22-s + (0.258 − 0.447i)23-s + (−0.863 − 1.49i)25-s + 0.116·26-s + ⋯

Functional equation

Λ(s)=(1323s/2 ΓC(s) L(s)=((−0.841+0.539i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.841 + 0.539i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1323s/2 ΓC(s+1/2) L(s)=((−0.841+0.539i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.841 + 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13231323    =    33⋅723^{3} \cdot 7^{2}
Sign: −0.841+0.539i-0.841 + 0.539i
Analytic conductor: 10.564210.5642
Root analytic conductor: 3.250263.25026
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1323(442,⋅)\chi_{1323} (442, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1323, ( :1/2), −0.841+0.539i)(2,\ 1323,\ (\ :1/2),\ -0.841 + 0.539i)

Particular Values

L(1)L(1) ≈\approx 1.7364538151.736453815
L(12)L(\frac12) ≈\approx 1.7364538151.736453815
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+(0.247+0.429i)T+(−1+1.73i)T2 1 + (0.247 + 0.429i)T + (-1 + 1.73i)T^{2}
5 1+(−1.84+3.19i)T+(−2.5−4.33i)T2 1 + (-1.84 + 3.19i)T + (-2.5 - 4.33i)T^{2}
11 1+(0.446+0.772i)T+(−5.5+9.52i)T2 1 + (0.446 + 0.772i)T + (-5.5 + 9.52i)T^{2}
13 1+(0.598−1.03i)T+(−6.5−11.2i)T2 1 + (0.598 - 1.03i)T + (-6.5 - 11.2i)T^{2}
17 1−0.249T+17T2 1 - 0.249T + 17T^{2}
19 1−2.80T+19T2 1 - 2.80T + 19T^{2}
23 1+(−1.23+2.14i)T+(−11.5−19.9i)T2 1 + (-1.23 + 2.14i)T + (-11.5 - 19.9i)T^{2}
29 1+(2.07+3.58i)T+(−14.5+25.1i)T2 1 + (2.07 + 3.58i)T + (-14.5 + 25.1i)T^{2}
31 1+(−1.79+3.10i)T+(−15.5−26.8i)T2 1 + (-1.79 + 3.10i)T + (-15.5 - 26.8i)T^{2}
37 1−4.73T+37T2 1 - 4.73T + 37T^{2}
41 1+(2.39−4.14i)T+(−20.5−35.5i)T2 1 + (2.39 - 4.14i)T + (-20.5 - 35.5i)T^{2}
43 1+(4.98+8.64i)T+(−21.5+37.2i)T2 1 + (4.98 + 8.64i)T + (-21.5 + 37.2i)T^{2}
47 1+(−5.08−8.81i)T+(−23.5+40.7i)T2 1 + (-5.08 - 8.81i)T + (-23.5 + 40.7i)T^{2}
53 1+9.88T+53T2 1 + 9.88T + 53T^{2}
59 1+(0.906−1.56i)T+(−29.5−51.0i)T2 1 + (0.906 - 1.56i)T + (-29.5 - 51.0i)T^{2}
61 1+(−5.40−9.35i)T+(−30.5+52.8i)T2 1 + (-5.40 - 9.35i)T + (-30.5 + 52.8i)T^{2}
67 1+(0.514−0.891i)T+(−33.5−58.0i)T2 1 + (0.514 - 0.891i)T + (-33.5 - 58.0i)T^{2}
71 1−4.94T+71T2 1 - 4.94T + 71T^{2}
73 1+1.83T+73T2 1 + 1.83T + 73T^{2}
79 1+(−0.899−1.55i)T+(−39.5+68.4i)T2 1 + (-0.899 - 1.55i)T + (-39.5 + 68.4i)T^{2}
83 1+(−6.16−10.6i)T+(−41.5+71.8i)T2 1 + (-6.16 - 10.6i)T + (-41.5 + 71.8i)T^{2}
89 1−2.40T+89T2 1 - 2.40T + 89T^{2}
97 1+(5.52+9.56i)T+(−48.5+84.0i)T2 1 + (5.52 + 9.56i)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

−9.479836882007481941305729611691, −8.751621706887782006751906787716, −7.83292216885667864700498707532, −6.63393456687661820688387494049, −5.80855653415181875511900011683, −5.22835250073408756055154499499, −4.32655778981553825276412871115, −2.72560135898294259269370286451, −1.70560738680489533992696614508, −0.74180193776828483859916356895, 1.95011694088265778649428404670, 2.93953088624929289907387728642, 3.52332301489832171316167343631, 5.10974088976400154308248342378, 6.10345509263089118589779515809, 6.78553947061436273875020493500, 7.38093341065729580026908899926, 8.134117690545274684537572739129, 9.249267974181079178881181953878, 9.923298378483550222664662040396

Graph of the ZZ-function along the critical line