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Properties

Label 2-147-7.2-c7-0-29
Degree 22
Conductor 147147
Sign 0.386+0.922i0.386 + 0.922i
Analytic cond. 45.920545.9205
Root an. cond. 6.776476.77647
Motivic weight 77
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (6.10 − 10.5i)2-s + (−13.5 − 23.3i)3-s + (−10.5 − 18.2i)4-s + (142. − 246. i)5-s − 329.·6-s + 1.30e3·8-s + (−364.5 + 631. i)9-s + (−1.73e3 − 3.01e3i)10-s + (3.64e3 + 6.31e3i)11-s + (−285. + 493. i)12-s + 8.51e3·13-s − 7.68e3·15-s + (9.32e3 − 1.61e4i)16-s + (1.67e4 + 2.89e4i)17-s + (4.45e3 + 7.70e3i)18-s + (4.45e3 − 7.71e3i)19-s + ⋯
L(s)  = 1  + (0.539 − 0.934i)2-s + (−0.288 − 0.499i)3-s + (−0.0825 − 0.142i)4-s + (0.509 − 0.882i)5-s − 0.623·6-s + 0.901·8-s + (−0.166 + 0.288i)9-s + (−0.549 − 0.952i)10-s + (0.825 + 1.42i)11-s + (−0.0476 + 0.0825i)12-s + 1.07·13-s − 0.588·15-s + (0.568 − 0.985i)16-s + (0.825 + 1.43i)17-s + (0.179 + 0.311i)18-s + (0.148 − 0.257i)19-s + ⋯

Functional equation

Λ(s)=(147s/2 ΓC(s) L(s)=((0.386+0.922i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(147s/2 ΓC(s+7/2) L(s)=((0.386+0.922i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+7/2) \, 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+0.922i0.386 + 0.922i
Analytic conductor: 45.920545.9205
Root analytic conductor: 6.776476.77647
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: χ147(79,⋅)\chi_{147} (79, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 147, ( :7/2), 0.386+0.922i)(2,\ 147,\ (\ :7/2),\ 0.386 + 0.922i)

Particular Values

L(4)L(4) ≈\approx 3.5863813363.586381336
L(12)L(\frac12) ≈\approx 3.5863813363.586381336
L(92)L(\frac{9}{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+(13.5+23.3i)T 1 + (13.5 + 23.3i)T
7 1 1
good2 1+(−6.10+10.5i)T+(−64−110.i)T2 1 + (-6.10 + 10.5i)T + (-64 - 110. i)T^{2}
5 1+(−142.+246.i)T+(−3.90e4−6.76e4i)T2 1 + (-142. + 246. i)T + (-3.90e4 - 6.76e4i)T^{2}
11 1+(−3.64e3−6.31e3i)T+(−9.74e6+1.68e7i)T2 1 + (-3.64e3 - 6.31e3i)T + (-9.74e6 + 1.68e7i)T^{2}
13 1−8.51e3T+6.27e7T2 1 - 8.51e3T + 6.27e7T^{2}
17 1+(−1.67e4−2.89e4i)T+(−2.05e8+3.55e8i)T2 1 + (-1.67e4 - 2.89e4i)T + (-2.05e8 + 3.55e8i)T^{2}
19 1+(−4.45e3+7.71e3i)T+(−4.46e8−7.74e8i)T2 1 + (-4.45e3 + 7.71e3i)T + (-4.46e8 - 7.74e8i)T^{2}
23 1+(2.72e4−4.71e4i)T+(−1.70e9−2.94e9i)T2 1 + (2.72e4 - 4.71e4i)T + (-1.70e9 - 2.94e9i)T^{2}
29 1+3.24e4T+1.72e10T2 1 + 3.24e4T + 1.72e10T^{2}
31 1+(−8.45e4−1.46e5i)T+(−1.37e10+2.38e10i)T2 1 + (-8.45e4 - 1.46e5i)T + (-1.37e10 + 2.38e10i)T^{2}
37 1+(4.96e4−8.59e4i)T+(−4.74e10−8.22e10i)T2 1 + (4.96e4 - 8.59e4i)T + (-4.74e10 - 8.22e10i)T^{2}
41 1−2.35e5T+1.94e11T2 1 - 2.35e5T + 1.94e11T^{2}
43 1+2.89e5T+2.71e11T2 1 + 2.89e5T + 2.71e11T^{2}
47 1+(−6.04e5+1.04e6i)T+(−2.53e11−4.38e11i)T2 1 + (-6.04e5 + 1.04e6i)T + (-2.53e11 - 4.38e11i)T^{2}
53 1+(−9.89e4−1.71e5i)T+(−5.87e11+1.01e12i)T2 1 + (-9.89e4 - 1.71e5i)T + (-5.87e11 + 1.01e12i)T^{2}
59 1+(1.11e6+1.92e6i)T+(−1.24e12+2.15e12i)T2 1 + (1.11e6 + 1.92e6i)T + (-1.24e12 + 2.15e12i)T^{2}
61 1+(5.30e5−9.18e5i)T+(−1.57e12−2.72e12i)T2 1 + (5.30e5 - 9.18e5i)T + (-1.57e12 - 2.72e12i)T^{2}
67 1+(1.87e6+3.24e6i)T+(−3.03e12+5.24e12i)T2 1 + (1.87e6 + 3.24e6i)T + (-3.03e12 + 5.24e12i)T^{2}
71 1+2.99e6T+9.09e12T2 1 + 2.99e6T + 9.09e12T^{2}
73 1+(3.28e6+5.69e6i)T+(−5.52e12+9.56e12i)T2 1 + (3.28e6 + 5.69e6i)T + (-5.52e12 + 9.56e12i)T^{2}
79 1+(3.54e6−6.14e6i)T+(−9.60e12−1.66e13i)T2 1 + (3.54e6 - 6.14e6i)T + (-9.60e12 - 1.66e13i)T^{2}
83 1+4.41e5T+2.71e13T2 1 + 4.41e5T + 2.71e13T^{2}
89 1+(5.92e6−1.02e7i)T+(−2.21e13−3.83e13i)T2 1 + (5.92e6 - 1.02e7i)T + (-2.21e13 - 3.83e13i)T^{2}
97 1−4.49e6T+8.07e13T2 1 - 4.49e6T + 8.07e13T^{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

−11.92695124740729032605461515622, −10.72758016254858843148500726313, −9.729485765703523612443861422641, −8.469929457853419244997646792815, −7.23010163012633003884373828935, −5.89404386556545927708035907722, −4.66742180985413836137642096944, −3.56011387611497406119922586134, −1.72372403479761316494863089348, −1.35619670645607308027649283627, 0.998571549406775535855489345688, 3.01464677831036009769022629561, 4.28430354580104729973503262554, 5.87706382077652038011646073457, 6.09230369089135261145654767301, 7.33489609802139204072823477900, 8.715117541148683485467210163993, 10.01117563979775456203459955483, 10.89741843927464523439439492620, 11.69912600178178114351582622306

Graph of the ZZ-function along the critical line