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Properties

Label 2-1323-63.25-c1-0-26
Degree 22
Conductor 13231323
Sign −0.678+0.734i-0.678 + 0.734i
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  − 1.34·2-s − 0.184·4-s + (1.26 − 2.19i)5-s + 2.94·8-s + (−1.70 + 2.95i)10-s + (0.233 + 0.405i)11-s + (−2.91 − 5.04i)13-s − 3.59·16-s + (1.93 − 3.35i)17-s + (1.09 + 1.89i)19-s + (−0.233 + 0.405i)20-s + (−0.315 − 0.545i)22-s + (−0.0530 + 0.0918i)23-s + (−0.705 − 1.22i)25-s + (3.92 + 6.79i)26-s + ⋯
L(s)  = 1  − 0.952·2-s − 0.0923·4-s + (0.566 − 0.980i)5-s + 1.04·8-s + (−0.539 + 0.934i)10-s + (0.0705 + 0.122i)11-s + (−0.807 − 1.39i)13-s − 0.899·16-s + (0.470 − 0.814i)17-s + (0.250 + 0.434i)19-s + (−0.0523 + 0.0906i)20-s + (−0.0672 − 0.116i)22-s + (−0.0110 + 0.0191i)23-s + (−0.141 − 0.244i)25-s + (0.769 + 1.33i)26-s + ⋯

Functional equation

Λ(s)=(1323s/2 ΓC(s) L(s)=((−0.678+0.734i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.678 + 0.734i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1323s/2 ΓC(s+1/2) L(s)=((−0.678+0.734i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.678 + 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13231323    =    33⋅723^{3} \cdot 7^{2}
Sign: −0.678+0.734i-0.678 + 0.734i
Analytic conductor: 10.564210.5642
Root analytic conductor: 3.250263.25026
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1323(802,⋅)\chi_{1323} (802, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1323, ( :1/2), −0.678+0.734i)(2,\ 1323,\ (\ :1/2),\ -0.678 + 0.734i)

Particular Values

L(1)L(1) ≈\approx 0.70605794360.7060579436
L(12)L(\frac12) ≈\approx 0.70605794360.7060579436
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+1.34T+2T2 1 + 1.34T + 2T^{2}
5 1+(−1.26+2.19i)T+(−2.5−4.33i)T2 1 + (-1.26 + 2.19i)T + (-2.5 - 4.33i)T^{2}
11 1+(−0.233−0.405i)T+(−5.5+9.52i)T2 1 + (-0.233 - 0.405i)T + (-5.5 + 9.52i)T^{2}
13 1+(2.91+5.04i)T+(−6.5+11.2i)T2 1 + (2.91 + 5.04i)T + (-6.5 + 11.2i)T^{2}
17 1+(−1.93+3.35i)T+(−8.5−14.7i)T2 1 + (-1.93 + 3.35i)T + (-8.5 - 14.7i)T^{2}
19 1+(−1.09−1.89i)T+(−9.5+16.4i)T2 1 + (-1.09 - 1.89i)T + (-9.5 + 16.4i)T^{2}
23 1+(0.0530−0.0918i)T+(−11.5−19.9i)T2 1 + (0.0530 - 0.0918i)T + (-11.5 - 19.9i)T^{2}
29 1+(−4.39+7.60i)T+(−14.5−25.1i)T2 1 + (-4.39 + 7.60i)T + (-14.5 - 25.1i)T^{2}
31 1+7.68T+31T2 1 + 7.68T + 31T^{2}
37 1+(−3.84−6.65i)T+(−18.5+32.0i)T2 1 + (-3.84 - 6.65i)T + (-18.5 + 32.0i)T^{2}
41 1+(1.11+1.92i)T+(−20.5+35.5i)T2 1 + (1.11 + 1.92i)T + (-20.5 + 35.5i)T^{2}
43 1+(0.613−1.06i)T+(−21.5−37.2i)T2 1 + (0.613 - 1.06i)T + (-21.5 - 37.2i)T^{2}
47 1−5.33T+47T2 1 - 5.33T + 47T^{2}
53 1+(0.358−0.620i)T+(−26.5−45.8i)T2 1 + (0.358 - 0.620i)T + (-26.5 - 45.8i)T^{2}
59 1+0.736T+59T2 1 + 0.736T + 59T^{2}
61 1−0.958T+61T2 1 - 0.958T + 61T^{2}
67 1+9.63T+67T2 1 + 9.63T + 67T^{2}
71 1+13.2T+71T2 1 + 13.2T + 71T^{2}
73 1+(−5.13+8.89i)T+(−36.5−63.2i)T2 1 + (-5.13 + 8.89i)T + (-36.5 - 63.2i)T^{2}
79 1+12.6T+79T2 1 + 12.6T + 79T^{2}
83 1+(1.36−2.36i)T+(−41.5−71.8i)T2 1 + (1.36 - 2.36i)T + (-41.5 - 71.8i)T^{2}
89 1+(4.05+7.02i)T+(−44.5+77.0i)T2 1 + (4.05 + 7.02i)T + (-44.5 + 77.0i)T^{2}
97 1+(−6.80+11.7i)T+(−48.5−84.0i)T2 1 + (-6.80 + 11.7i)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

−9.386736093297137183208907528362, −8.632920556891456090505401288560, −7.85003893481137941710526620717, −7.30465567411923526402838435651, −5.84431803504516293134210128832, −5.15647210108345494806655841704, −4.37733122918936761059266255226, −2.88553098963996835285390837704, −1.50691906677424877124384919067, −0.43651933964689686108467198923, 1.49376423775150891738678551928, 2.53019707865550335147930614360, 3.84639525545306640528622805239, 4.86380285581696211941548942749, 5.96580661870092822463994557396, 7.01159818697596216983890284844, 7.34765527993913817589122741752, 8.546419604787823953684772182599, 9.189273954470415373296145774940, 9.826699419650117911064212539462

Graph of the ZZ-function along the critical line