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Properties

Label 2-1350-9.7-c1-0-17
Degree 22
Conductor 13501350
Sign −0.998+0.0561i-0.998 + 0.0561i
Analytic cond. 10.779810.7798
Root an. cond. 3.283263.28326
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  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (1.68 − 2.92i)7-s − 0.999·8-s + (−2.18 + 3.78i)11-s + (−3.37 − 5.84i)13-s + (−1.68 − 2.92i)14-s + (−0.5 + 0.866i)16-s − 1.62·17-s − 2.37·19-s + (2.18 + 3.78i)22-s + (−0.686 − 1.18i)23-s − 6.74·26-s − 3.37·28-s + (0.686 − 1.18i)29-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.637 − 1.10i)7-s − 0.353·8-s + (−0.659 + 1.14i)11-s + (−0.935 − 1.61i)13-s + (−0.450 − 0.780i)14-s + (−0.125 + 0.216i)16-s − 0.394·17-s − 0.544·19-s + (0.466 + 0.807i)22-s + (−0.143 − 0.247i)23-s − 1.32·26-s − 0.637·28-s + (0.127 − 0.220i)29-s + ⋯

Functional equation

Λ(s)=(1350s/2 ΓC(s) L(s)=((−0.998+0.0561i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0561i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1350s/2 ΓC(s+1/2) L(s)=((−0.998+0.0561i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0561i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13501350    =    2⋅33⋅522 \cdot 3^{3} \cdot 5^{2}
Sign: −0.998+0.0561i-0.998 + 0.0561i
Analytic conductor: 10.779810.7798
Root analytic conductor: 3.283263.28326
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1350(451,⋅)\chi_{1350} (451, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1350, ( :1/2), −0.998+0.0561i)(2,\ 1350,\ (\ :1/2),\ -0.998 + 0.0561i)

Particular Values

L(1)L(1) ≈\approx 1.1554493981.155449398
L(12)L(\frac12) ≈\approx 1.1554493981.155449398
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+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
3 1 1
5 1 1
good7 1+(−1.68+2.92i)T+(−3.5−6.06i)T2 1 + (-1.68 + 2.92i)T + (-3.5 - 6.06i)T^{2}
11 1+(2.18−3.78i)T+(−5.5−9.52i)T2 1 + (2.18 - 3.78i)T + (-5.5 - 9.52i)T^{2}
13 1+(3.37+5.84i)T+(−6.5+11.2i)T2 1 + (3.37 + 5.84i)T + (-6.5 + 11.2i)T^{2}
17 1+1.62T+17T2 1 + 1.62T + 17T^{2}
19 1+2.37T+19T2 1 + 2.37T + 19T^{2}
23 1+(0.686+1.18i)T+(−11.5+19.9i)T2 1 + (0.686 + 1.18i)T + (-11.5 + 19.9i)T^{2}
29 1+(−0.686+1.18i)T+(−14.5−25.1i)T2 1 + (-0.686 + 1.18i)T + (-14.5 - 25.1i)T^{2}
31 1+(2.37+4.10i)T+(−15.5+26.8i)T2 1 + (2.37 + 4.10i)T + (-15.5 + 26.8i)T^{2}
37 1−4T+37T2 1 - 4T + 37T^{2}
41 1+(1.5+2.59i)T+(−20.5+35.5i)T2 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2}
43 1+(2.81−4.87i)T+(−21.5−37.2i)T2 1 + (2.81 - 4.87i)T + (-21.5 - 37.2i)T^{2}
47 1+(−3.68+6.38i)T+(−23.5−40.7i)T2 1 + (-3.68 + 6.38i)T + (-23.5 - 40.7i)T^{2}
53 1+11.4T+53T2 1 + 11.4T + 53T^{2}
59 1+(−2.18−3.78i)T+(−29.5+51.0i)T2 1 + (-2.18 - 3.78i)T + (-29.5 + 51.0i)T^{2}
61 1+(−4.05+7.02i)T+(−30.5−52.8i)T2 1 + (-4.05 + 7.02i)T + (-30.5 - 52.8i)T^{2}
67 1+(3.5+6.06i)T+(−33.5+58.0i)T2 1 + (3.5 + 6.06i)T + (-33.5 + 58.0i)T^{2}
71 1−6T+71T2 1 - 6T + 71T^{2}
73 1+3.11T+73T2 1 + 3.11T + 73T^{2}
79 1+(1−1.73i)T+(−39.5−68.4i)T2 1 + (1 - 1.73i)T + (-39.5 - 68.4i)T^{2}
83 1+(3.68−6.38i)T+(−41.5−71.8i)T2 1 + (3.68 - 6.38i)T + (-41.5 - 71.8i)T^{2}
89 1+16.1T+89T2 1 + 16.1T + 89T^{2}
97 1+(4.18−7.25i)T+(−48.5−84.0i)T2 1 + (4.18 - 7.25i)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

−9.594793198102994899655294353226, −8.208077192333327158054421164463, −7.67703162144427077941433231123, −6.85939501111415035919907215913, −5.58035663198225969691658068112, −4.77247838521949474450934730247, −4.16250271313971930805068668779, −2.89505671910574810782854984780, −1.91976876368333965946803393861, −0.39325568901143939673284207071, 1.96516205169217768083173587473, 2.94958009748604298446120992469, 4.28606288454581634274843902220, 5.05287122772067412600783394698, 5.82101939346536012136074021965, 6.64360564819188993231464793411, 7.53836564486828907623381113602, 8.505697753750577879836377048411, 8.866731704692558606784359831743, 9.786119410953303222133753381929

Graph of the ZZ-function along the critical line