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Properties

Label 2-675-1.1-c3-0-20
Degree 22
Conductor 675675
Sign 11
Analytic cond. 39.826239.8262
Root an. cond. 6.310806.31080
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.90·2-s + 16.1·4-s + 2.10·7-s − 39.7·8-s + 40.3·11-s + 61.7·13-s − 10.3·14-s + 66.5·16-s − 99.2·17-s + 134.·19-s − 197.·22-s − 76.4·23-s − 303.·26-s + 33.8·28-s + 236.·29-s + 243.·31-s − 8.27·32-s + 487.·34-s + 57.4·37-s − 660.·38-s − 411.·41-s + 102.·43-s + 649.·44-s + 375.·46-s + 260.·47-s − 338.·49-s + 994.·52-s + ⋯
L(s)  = 1  − 1.73·2-s + 2.01·4-s + 0.113·7-s − 1.75·8-s + 1.10·11-s + 1.31·13-s − 0.197·14-s + 1.03·16-s − 1.41·17-s + 1.62·19-s − 1.91·22-s − 0.693·23-s − 2.28·26-s + 0.228·28-s + 1.51·29-s + 1.41·31-s − 0.0457·32-s + 2.45·34-s + 0.255·37-s − 2.81·38-s − 1.56·41-s + 0.364·43-s + 2.22·44-s + 1.20·46-s + 0.808·47-s − 0.987·49-s + 2.65·52-s + ⋯

Functional equation

Λ(s)=(675s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(675s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 675675    =    33⋅523^{3} \cdot 5^{2}
Sign: 11
Analytic conductor: 39.826239.8262
Root analytic conductor: 6.310806.31080
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 675, ( :3/2), 1)(2,\ 675,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.0119511631.011951163
L(12)L(\frac12) ≈\approx 1.0119511631.011951163
L(52)L(\frac{5}{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
5 1 1
good2 1+4.90T+8T2 1 + 4.90T + 8T^{2}
7 1−2.10T+343T2 1 - 2.10T + 343T^{2}
11 1−40.3T+1.33e3T2 1 - 40.3T + 1.33e3T^{2}
13 1−61.7T+2.19e3T2 1 - 61.7T + 2.19e3T^{2}
17 1+99.2T+4.91e3T2 1 + 99.2T + 4.91e3T^{2}
19 1−134.T+6.85e3T2 1 - 134.T + 6.85e3T^{2}
23 1+76.4T+1.21e4T2 1 + 76.4T + 1.21e4T^{2}
29 1−236.T+2.43e4T2 1 - 236.T + 2.43e4T^{2}
31 1−243.T+2.97e4T2 1 - 243.T + 2.97e4T^{2}
37 1−57.4T+5.06e4T2 1 - 57.4T + 5.06e4T^{2}
41 1+411.T+6.89e4T2 1 + 411.T + 6.89e4T^{2}
43 1−102.T+7.95e4T2 1 - 102.T + 7.95e4T^{2}
47 1−260.T+1.03e5T2 1 - 260.T + 1.03e5T^{2}
53 1+75.4T+1.48e5T2 1 + 75.4T + 1.48e5T^{2}
59 1+39.2T+2.05e5T2 1 + 39.2T + 2.05e5T^{2}
61 1+675.T+2.26e5T2 1 + 675.T + 2.26e5T^{2}
67 1+601.T+3.00e5T2 1 + 601.T + 3.00e5T^{2}
71 1−222.T+3.57e5T2 1 - 222.T + 3.57e5T^{2}
73 1+297.T+3.89e5T2 1 + 297.T + 3.89e5T^{2}
79 1+95.6T+4.93e5T2 1 + 95.6T + 4.93e5T^{2}
83 1−3.08T+5.71e5T2 1 - 3.08T + 5.71e5T^{2}
89 1−1.34e3T+7.04e5T2 1 - 1.34e3T + 7.04e5T^{2}
97 1−1.48e3T+9.12e5T2 1 - 1.48e3T + 9.12e5T^{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

−9.966209534330282307583862733229, −9.094948017171529039436944893565, −8.586915611400940022711131782860, −7.76454627256167357763963981239, −6.68013598294086809194357647346, −6.20036895870315709240743655571, −4.49449715127709360659026504732, −3.10380614250885153935170072674, −1.68312266626246378905505047641, −0.810388981678696396101618511468, 0.810388981678696396101618511468, 1.68312266626246378905505047641, 3.10380614250885153935170072674, 4.49449715127709360659026504732, 6.20036895870315709240743655571, 6.68013598294086809194357647346, 7.76454627256167357763963981239, 8.586915611400940022711131782860, 9.094948017171529039436944893565, 9.966209534330282307583862733229

Graph of the ZZ-function along the critical line