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Properties

Label 2-147-7.4-c5-0-27
Degree 22
Conductor 147147
Sign −0.386+0.922i-0.386 + 0.922i
Analytic cond. 23.576423.5764
Root an. cond. 4.855554.85555
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3 + 5.19i)2-s + (4.5 − 7.79i)3-s + (−2 + 3.46i)4-s + (−39 − 67.5i)5-s + 54·6-s + 168·8-s + (−40.5 − 70.1i)9-s + (234 − 405. i)10-s + (−222 + 384. i)11-s + (18.0 + 31.1i)12-s − 442·13-s − 702·15-s + (568 + 983. i)16-s + (63 − 109. i)17-s + (243 − 420. i)18-s + (−1.34e3 − 2.32e3i)19-s + ⋯
L(s)  = 1  + (0.530 + 0.918i)2-s + (0.288 − 0.499i)3-s + (−0.0625 + 0.108i)4-s + (−0.697 − 1.20i)5-s + 0.612·6-s + 0.928·8-s + (−0.166 − 0.288i)9-s + (0.739 − 1.28i)10-s + (−0.553 + 0.958i)11-s + (0.0360 + 0.0625i)12-s − 0.725·13-s − 0.805·15-s + (0.554 + 0.960i)16-s + (0.0528 − 0.0915i)17-s + (0.176 − 0.306i)18-s + (−0.852 − 1.47i)19-s + ⋯

Functional equation

Λ(s)=(147s/2 ΓC(s) L(s)=((−0.386+0.922i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(147s/2 ΓC(s+5/2) L(s)=((−0.386+0.922i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 147147    =    3⋅723 \cdot 7^{2}
Sign: −0.386+0.922i-0.386 + 0.922i
Analytic conductor: 23.576423.5764
Root analytic conductor: 4.855554.85555
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ147(67,⋅)\chi_{147} (67, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 147, ( :5/2), −0.386+0.922i)(2,\ 147,\ (\ :5/2),\ -0.386 + 0.922i)

Particular Values

L(3)L(3) ≈\approx 1.4611012611.461101261
L(12)L(\frac12) ≈\approx 1.4611012611.461101261
L(72)L(\frac{7}{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+(−4.5+7.79i)T 1 + (-4.5 + 7.79i)T
7 1 1
good2 1+(−3−5.19i)T+(−16+27.7i)T2 1 + (-3 - 5.19i)T + (-16 + 27.7i)T^{2}
5 1+(39+67.5i)T+(−1.56e3+2.70e3i)T2 1 + (39 + 67.5i)T + (-1.56e3 + 2.70e3i)T^{2}
11 1+(222−384.i)T+(−8.05e4−1.39e5i)T2 1 + (222 - 384. i)T + (-8.05e4 - 1.39e5i)T^{2}
13 1+442T+3.71e5T2 1 + 442T + 3.71e5T^{2}
17 1+(−63+109.i)T+(−7.09e5−1.22e6i)T2 1 + (-63 + 109. i)T + (-7.09e5 - 1.22e6i)T^{2}
19 1+(1.34e3+2.32e3i)T+(−1.23e6+2.14e6i)T2 1 + (1.34e3 + 2.32e3i)T + (-1.23e6 + 2.14e6i)T^{2}
23 1+(2.10e3+3.63e3i)T+(−3.21e6+5.57e6i)T2 1 + (2.10e3 + 3.63e3i)T + (-3.21e6 + 5.57e6i)T^{2}
29 1+5.44e3T+2.05e7T2 1 + 5.44e3T + 2.05e7T^{2}
31 1+(40−69.2i)T+(−1.43e7−2.47e7i)T2 1 + (40 - 69.2i)T + (-1.43e7 - 2.47e7i)T^{2}
37 1+(−2.71e3−4.70e3i)T+(−3.46e7+6.00e7i)T2 1 + (-2.71e3 - 4.70e3i)T + (-3.46e7 + 6.00e7i)T^{2}
41 1−7.96e3T+1.15e8T2 1 - 7.96e3T + 1.15e8T^{2}
43 1+1.15e4T+1.47e8T2 1 + 1.15e4T + 1.47e8T^{2}
47 1+(−6.96e3−1.20e4i)T+(−1.14e8+1.98e8i)T2 1 + (-6.96e3 - 1.20e4i)T + (-1.14e8 + 1.98e8i)T^{2}
53 1+(−4.79e3+8.30e3i)T+(−2.09e8−3.62e8i)T2 1 + (-4.79e3 + 8.30e3i)T + (-2.09e8 - 3.62e8i)T^{2}
59 1+(1.37e4−2.38e4i)T+(−3.57e8−6.19e8i)T2 1 + (1.37e4 - 2.38e4i)T + (-3.57e8 - 6.19e8i)T^{2}
61 1+(2.47e4+4.28e4i)T+(−4.22e8+7.31e8i)T2 1 + (2.47e4 + 4.28e4i)T + (-4.22e8 + 7.31e8i)T^{2}
67 1+(−2.96e4+5.14e4i)T+(−6.75e8−1.16e9i)T2 1 + (-2.96e4 + 5.14e4i)T + (-6.75e8 - 1.16e9i)T^{2}
71 1−3.20e4T+1.80e9T2 1 - 3.20e4T + 1.80e9T^{2}
73 1+(−3.09e4+5.35e4i)T+(−1.03e9−1.79e9i)T2 1 + (-3.09e4 + 5.35e4i)T + (-1.03e9 - 1.79e9i)T^{2}
79 1+(−3.28e4−5.69e4i)T+(−1.53e9+2.66e9i)T2 1 + (-3.28e4 - 5.69e4i)T + (-1.53e9 + 2.66e9i)T^{2}
83 1−4.01e4T+3.93e9T2 1 - 4.01e4T + 3.93e9T^{2}
89 1+(−3.98e3−6.90e3i)T+(−2.79e9+4.83e9i)T2 1 + (-3.98e3 - 6.90e3i)T + (-2.79e9 + 4.83e9i)T^{2}
97 1+1.43e5T+8.58e9T2 1 + 1.43e5T + 8.58e9T^{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

−12.37267942221707701866710971737, −10.92813965320586952857455138305, −9.504787738818300343770048651040, −8.268769414663246479157885117185, −7.52306568955418276409634218132, −6.50131624677641011352285160093, −5.00084021470008397935389571515, −4.39714423639403199621530903871, −2.12192349625375215475327525950, −0.37103823368060179196664231475, 2.18573393359572254728069463825, 3.35684217607051297998196800219, 3.97110970669868766338939510135, 5.67061899817867306119845861452, 7.37058665093314473129618239347, 8.110807781561089577122431050035, 9.866477786776292022879285271579, 10.70787242846637690066120869274, 11.33722347091717048290249089610, 12.25735097790308652579408349638

Graph of the ZZ-function along the critical line