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Properties

Label 2-161-7.4-c3-0-0
Degree 22
Conductor 161161
Sign −0.948−0.316i-0.948 - 0.316i
Analytic cond. 9.499309.49930
Root an. cond. 3.082093.08209
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.59 − 4.49i)2-s + (−3.54 + 6.13i)3-s + (−9.44 + 16.3i)4-s + (2.19 + 3.80i)5-s + 36.7·6-s + (−4.74 + 17.9i)7-s + 56.4·8-s + (−11.5 − 20.0i)9-s + (11.4 − 19.7i)10-s + (4.40 − 7.63i)11-s + (−66.9 − 115. i)12-s + 4.71·13-s + (92.6 − 25.1i)14-s − 31.1·15-s + (−70.8 − 122. i)16-s + (−66.9 + 115. i)17-s + ⋯
L(s)  = 1  + (−0.916 − 1.58i)2-s + (−0.681 + 1.18i)3-s + (−1.18 + 2.04i)4-s + (0.196 + 0.340i)5-s + 2.49·6-s + (−0.255 + 0.966i)7-s + 2.49·8-s + (−0.429 − 0.743i)9-s + (0.360 − 0.624i)10-s + (0.120 − 0.209i)11-s + (−1.60 − 2.78i)12-s + 0.100·13-s + (1.76 − 0.479i)14-s − 0.536·15-s + (−1.10 − 1.91i)16-s + (−0.954 + 1.65i)17-s + ⋯

Functional equation

Λ(s)=(161s/2 ΓC(s) L(s)=((−0.948−0.316i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(161s/2 ΓC(s+3/2) L(s)=((−0.948−0.316i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 161161    =    7⋅237 \cdot 23
Sign: −0.948−0.316i-0.948 - 0.316i
Analytic conductor: 9.499309.49930
Root analytic conductor: 3.082093.08209
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ161(116,⋅)\chi_{161} (116, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 161, ( :3/2), −0.948−0.316i)(2,\ 161,\ (\ :3/2),\ -0.948 - 0.316i)

Particular Values

L(2)L(2) ≈\approx 0.0221339+0.136185i0.0221339 + 0.136185i
L(12)L(\frac12) ≈\approx 0.0221339+0.136185i0.0221339 + 0.136185i
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)
bad7 1+(4.74−17.9i)T 1 + (4.74 - 17.9i)T
23 1+(−11.5−19.9i)T 1 + (-11.5 - 19.9i)T
good2 1+(2.59+4.49i)T+(−4+6.92i)T2 1 + (2.59 + 4.49i)T + (-4 + 6.92i)T^{2}
3 1+(3.54−6.13i)T+(−13.5−23.3i)T2 1 + (3.54 - 6.13i)T + (-13.5 - 23.3i)T^{2}
5 1+(−2.19−3.80i)T+(−62.5+108.i)T2 1 + (-2.19 - 3.80i)T + (-62.5 + 108. i)T^{2}
11 1+(−4.40+7.63i)T+(−665.5−1.15e3i)T2 1 + (-4.40 + 7.63i)T + (-665.5 - 1.15e3i)T^{2}
13 1−4.71T+2.19e3T2 1 - 4.71T + 2.19e3T^{2}
17 1+(66.9−115.i)T+(−2.45e3−4.25e3i)T2 1 + (66.9 - 115. i)T + (-2.45e3 - 4.25e3i)T^{2}
19 1+(48.9+84.6i)T+(−3.42e3+5.94e3i)T2 1 + (48.9 + 84.6i)T + (-3.42e3 + 5.94e3i)T^{2}
29 1−103.T+2.43e4T2 1 - 103.T + 2.43e4T^{2}
31 1+(27.6−47.8i)T+(−1.48e4−2.57e4i)T2 1 + (27.6 - 47.8i)T + (-1.48e4 - 2.57e4i)T^{2}
37 1+(74.5+129.i)T+(−2.53e4+4.38e4i)T2 1 + (74.5 + 129. i)T + (-2.53e4 + 4.38e4i)T^{2}
41 1+369.T+6.89e4T2 1 + 369.T + 6.89e4T^{2}
43 1+34.6T+7.95e4T2 1 + 34.6T + 7.95e4T^{2}
47 1+(−78.9−136.i)T+(−5.19e4+8.99e4i)T2 1 + (-78.9 - 136. i)T + (-5.19e4 + 8.99e4i)T^{2}
53 1+(−381.+660.i)T+(−7.44e4−1.28e5i)T2 1 + (-381. + 660. i)T + (-7.44e4 - 1.28e5i)T^{2}
59 1+(279.−483.i)T+(−1.02e5−1.77e5i)T2 1 + (279. - 483. i)T + (-1.02e5 - 1.77e5i)T^{2}
61 1+(453.+786.i)T+(−1.13e5+1.96e5i)T2 1 + (453. + 786. i)T + (-1.13e5 + 1.96e5i)T^{2}
67 1+(50.8−87.9i)T+(−1.50e5−2.60e5i)T2 1 + (50.8 - 87.9i)T + (-1.50e5 - 2.60e5i)T^{2}
71 1+430.T+3.57e5T2 1 + 430.T + 3.57e5T^{2}
73 1+(−230.+399.i)T+(−1.94e5−3.36e5i)T2 1 + (-230. + 399. i)T + (-1.94e5 - 3.36e5i)T^{2}
79 1+(−153.−265.i)T+(−2.46e5+4.26e5i)T2 1 + (-153. - 265. i)T + (-2.46e5 + 4.26e5i)T^{2}
83 1−42.8T+5.71e5T2 1 - 42.8T + 5.71e5T^{2}
89 1+(114.+199.i)T+(−3.52e5+6.10e5i)T2 1 + (114. + 199. i)T + (-3.52e5 + 6.10e5i)T^{2}
97 1−678.T+9.12e5T2 1 - 678.T + 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

−12.39967922184705208763244403618, −11.45498780687599715149846870782, −10.70877754144381023518454982715, −10.18699420169044939592345741178, −9.078132512239341226818382888970, −8.499169567462399196424419302199, −6.33916266723967675368999076772, −4.74259633517986736307852167028, −3.53549485911125709539297631958, −2.16337426600112317519689004612, 0.10551306270422632775040078033, 1.27401740972791194081424457866, 4.75181416801699845294420809244, 5.98596579853347576740230887508, 6.90770405606662936522609292491, 7.34868700025308700492956017285, 8.546817443962706853864920394942, 9.620390941888289279133649051081, 10.69965823131721211360176085496, 12.06202178504834168133024065855

Graph of the ZZ-function along the critical line