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Properties

Label 2-162-9.2-c4-0-14
Degree 22
Conductor 162162
Sign −0.996+0.0871i-0.996 + 0.0871i
Analytic cond. 16.745916.7459
Root an. cond. 4.092174.09217
Motivic weight 44
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.44 − 1.41i)2-s + (3.99 + 6.92i)4-s + (7.97 − 4.60i)5-s + (−27.2 + 47.2i)7-s − 22.6i·8-s − 26.0·10-s + (34.3 + 19.8i)11-s + (−107. − 186. i)13-s + (133. − 77.1i)14-s + (−32.0 + 55.4i)16-s + 28.2i·17-s − 254.·19-s + (63.7 + 36.8i)20-s + (−56.0 − 97.0i)22-s + (801. − 462. i)23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (0.318 − 0.184i)5-s + (−0.556 + 0.964i)7-s − 0.353i·8-s − 0.260·10-s + (0.283 + 0.163i)11-s + (−0.638 − 1.10i)13-s + (0.681 − 0.393i)14-s + (−0.125 + 0.216i)16-s + 0.0976i·17-s − 0.704·19-s + (0.159 + 0.0920i)20-s + (−0.115 − 0.200i)22-s + (1.51 − 0.875i)23-s + ⋯

Functional equation

Λ(s)=(162s/2 ΓC(s) L(s)=((−0.996+0.0871i) Λ‾(5−s)\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0871i)\, \overline{\Lambda}(5-s) \end{aligned}
Λ(s)=(162s/2 ΓC(s+2) L(s)=((−0.996+0.0871i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0871i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 162162    =    2⋅342 \cdot 3^{4}
Sign: −0.996+0.0871i-0.996 + 0.0871i
Analytic conductor: 16.745916.7459
Root analytic conductor: 4.092174.09217
Motivic weight: 44
Rational: no
Arithmetic: yes
Character: χ162(107,⋅)\chi_{162} (107, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 162, ( :2), −0.996+0.0871i)(2,\ 162,\ (\ :2),\ -0.996 + 0.0871i)

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 0.20089580670.2008958067
L(12)L(\frac12) ≈\approx 0.20089580670.2008958067
L(3)L(3) 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+(2.44+1.41i)T 1 + (2.44 + 1.41i)T
3 1 1
good5 1+(−7.97+4.60i)T+(312.5−541.i)T2 1 + (-7.97 + 4.60i)T + (312.5 - 541. i)T^{2}
7 1+(27.2−47.2i)T+(−1.20e3−2.07e3i)T2 1 + (27.2 - 47.2i)T + (-1.20e3 - 2.07e3i)T^{2}
11 1+(−34.3−19.8i)T+(7.32e3+1.26e4i)T2 1 + (-34.3 - 19.8i)T + (7.32e3 + 1.26e4i)T^{2}
13 1+(107.+186.i)T+(−1.42e4+2.47e4i)T2 1 + (107. + 186. i)T + (-1.42e4 + 2.47e4i)T^{2}
17 1−28.2iT−8.35e4T2 1 - 28.2iT - 8.35e4T^{2}
19 1+254.T+1.30e5T2 1 + 254.T + 1.30e5T^{2}
23 1+(−801.+462.i)T+(1.39e5−2.42e5i)T2 1 + (-801. + 462. i)T + (1.39e5 - 2.42e5i)T^{2}
29 1+(1.16e3+672.i)T+(3.53e5+6.12e5i)T2 1 + (1.16e3 + 672. i)T + (3.53e5 + 6.12e5i)T^{2}
31 1+(588.+1.01e3i)T+(−4.61e5+7.99e5i)T2 1 + (588. + 1.01e3i)T + (-4.61e5 + 7.99e5i)T^{2}
37 1+2.34e3T+1.87e6T2 1 + 2.34e3T + 1.87e6T^{2}
41 1+(1.25e3−726.i)T+(1.41e6−2.44e6i)T2 1 + (1.25e3 - 726. i)T + (1.41e6 - 2.44e6i)T^{2}
43 1+(837.−1.45e3i)T+(−1.70e6−2.96e6i)T2 1 + (837. - 1.45e3i)T + (-1.70e6 - 2.96e6i)T^{2}
47 1+(2.87e3+1.66e3i)T+(2.43e6+4.22e6i)T2 1 + (2.87e3 + 1.66e3i)T + (2.43e6 + 4.22e6i)T^{2}
53 1−1.74e3iT−7.89e6T2 1 - 1.74e3iT - 7.89e6T^{2}
59 1+(709.−409.i)T+(6.05e6−1.04e7i)T2 1 + (709. - 409. i)T + (6.05e6 - 1.04e7i)T^{2}
61 1+(393.−682.i)T+(−6.92e6−1.19e7i)T2 1 + (393. - 682. i)T + (-6.92e6 - 1.19e7i)T^{2}
67 1+(1.42e3+2.47e3i)T+(−1.00e7+1.74e7i)T2 1 + (1.42e3 + 2.47e3i)T + (-1.00e7 + 1.74e7i)T^{2}
71 1+7.90e3iT−2.54e7T2 1 + 7.90e3iT - 2.54e7T^{2}
73 1−7.11e3T+2.83e7T2 1 - 7.11e3T + 2.83e7T^{2}
79 1+(−1.73e3+3.00e3i)T+(−1.94e7−3.37e7i)T2 1 + (-1.73e3 + 3.00e3i)T + (-1.94e7 - 3.37e7i)T^{2}
83 1+(996.+575.i)T+(2.37e7+4.11e7i)T2 1 + (996. + 575. i)T + (2.37e7 + 4.11e7i)T^{2}
89 1+1.07e4iT−6.27e7T2 1 + 1.07e4iT - 6.27e7T^{2}
97 1+(4.98e3−8.63e3i)T+(−4.42e7−7.66e7i)T2 1 + (4.98e3 - 8.63e3i)T + (-4.42e7 - 7.66e7i)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

−11.67122680763188498093976212416, −10.56194184894103124609509602276, −9.547649152978127748430078526344, −8.861558154636838606347031689796, −7.66359067579579335905545687218, −6.35288848028217649174576634986, −5.15669432275966114804094319915, −3.28135416269961450335083021817, −2.01184541473869320880390228628, −0.088888185955104982266063468310, 1.70035321516695241219496604682, 3.58845529223399943831746453426, 5.17391955478848289890213021080, 6.76450054630091061041283340742, 7.12085325586985075994172141967, 8.709203332036485648526146100805, 9.579456297712314060416143514952, 10.48302867601768463882267130022, 11.41046220677126944928942752632, 12.72376247643983986350725226299

Graph of the ZZ-function along the critical line