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Properties

Label 2-3240-360.149-c0-0-3
Degree 22
Conductor 32403240
Sign −0.984+0.173i-0.984 + 0.173i
Analytic cond. 1.616971.61697
Root an. cond. 1.271601.27160
Motivic weight 00
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 + (−0.5 + 0.866i)5-s − 0.999·8-s − 0.999·10-s + (0.5 + 0.866i)11-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)16-s + 17-s + (−0.499 − 0.866i)20-s + (−0.499 + 0.866i)22-s + (−0.5 + 0.866i)23-s + (−0.499 − 0.866i)25-s − 0.999·26-s + (0.5 + 0.866i)29-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (−0.5 + 0.866i)5-s − 0.999·8-s − 0.999·10-s + (0.5 + 0.866i)11-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)16-s + 17-s + (−0.499 − 0.866i)20-s + (−0.499 + 0.866i)22-s + (−0.5 + 0.866i)23-s + (−0.499 − 0.866i)25-s − 0.999·26-s + (0.5 + 0.866i)29-s + ⋯

Functional equation

Λ(s)=(3240s/2 ΓC(s) L(s)=((−0.984+0.173i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3240s/2 ΓC(s) L(s)=((−0.984+0.173i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32403240    =    23⋅34⋅52^{3} \cdot 3^{4} \cdot 5
Sign: −0.984+0.173i-0.984 + 0.173i
Analytic conductor: 1.616971.61697
Root analytic conductor: 1.271601.27160
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3240(1349,⋅)\chi_{3240} (1349, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3240, ( :0), −0.984+0.173i)(2,\ 3240,\ (\ :0),\ -0.984 + 0.173i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.1238718131.123871813
L(12)L(\frac12) ≈\approx 1.1238718131.123871813
L(1)L(1) 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+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
good7 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
11 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
13 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
17 1−T+T2 1 - T + T^{2}
19 1−T2 1 - T^{2}
23 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
29 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
31 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
37 1+2T+T2 1 + 2T + T^{2}
41 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
43 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
47 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
53 1−T2 1 - T^{2}
59 1+(1−1.73i)T+(−0.5−0.866i)T2 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2}
61 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
67 1+(−1+1.73i)T+(−0.5−0.866i)T2 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2}
71 1−T2 1 - T^{2}
73 1−T2 1 - T^{2}
79 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
83 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
89 1−T2 1 - T^{2}
97 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)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

−9.151249437841922681145230368014, −8.199630352896947043010679864993, −7.51884823665551572020554574559, −6.96776127452110724148730143219, −6.44525647106268202295484074576, −5.45326559416163228544252583361, −4.64231365639207788635566454689, −3.82692719228846909837839356870, −3.18297503559046341730402291264, −1.95712690178876785351690882242, 0.57968034872883906694187882052, 1.64134411004416809183873858310, 3.01539536230056019024095962902, 3.57046981248565124439813858206, 4.56164372473304840249343175666, 5.17807948522031802136723805143, 5.90736231249379593415394111778, 6.78620523954121927844545648154, 8.145273314305044598185963799598, 8.328841432101310499782599272380

Graph of the ZZ-function along the critical line