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Properties

Label 2-170-85.57-c1-0-3
Degree 22
Conductor 170170
Sign 0.976−0.216i0.976 - 0.216i
Analytic cond. 1.357451.35745
Root an. cond. 1.165091.16509
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.382 − 0.923i)2-s + (−0.189 + 0.283i)3-s + (−0.707 + 0.707i)4-s + (1.79 + 1.32i)5-s + (0.334 + 0.0666i)6-s + (−0.701 + 3.52i)7-s + (0.923 + 0.382i)8-s + (1.10 + 2.66i)9-s + (0.540 − 2.16i)10-s + (0.940 − 4.72i)11-s + (−0.0666 − 0.334i)12-s − 1.94i·13-s + (3.52 − 0.701i)14-s + (−0.718 + 0.258i)15-s − i·16-s + (1.93 + 3.64i)17-s + ⋯
L(s)  = 1  + (−0.270 − 0.653i)2-s + (−0.109 + 0.163i)3-s + (−0.353 + 0.353i)4-s + (0.804 + 0.594i)5-s + (0.136 + 0.0271i)6-s + (−0.264 + 1.33i)7-s + (0.326 + 0.135i)8-s + (0.367 + 0.887i)9-s + (0.170 − 0.686i)10-s + (0.283 − 1.42i)11-s + (−0.0192 − 0.0966i)12-s − 0.538i·13-s + (0.941 − 0.187i)14-s + (−0.185 + 0.0666i)15-s − 0.250i·16-s + (0.468 + 0.883i)17-s + ⋯

Functional equation

Λ(s)=(170s/2 ΓC(s) L(s)=((0.976−0.216i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.216i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(170s/2 ΓC(s+1/2) L(s)=((0.976−0.216i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.976 - 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 170170    =    2⋅5⋅172 \cdot 5 \cdot 17
Sign: 0.976−0.216i0.976 - 0.216i
Analytic conductor: 1.357451.35745
Root analytic conductor: 1.165091.16509
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ170(57,⋅)\chi_{170} (57, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 170, ( :1/2), 0.976−0.216i)(2,\ 170,\ (\ :1/2),\ 0.976 - 0.216i)

Particular Values

L(1)L(1) ≈\approx 1.05088+0.115274i1.05088 + 0.115274i
L(12)L(\frac12) ≈\approx 1.05088+0.115274i1.05088 + 0.115274i
L(32)L(\frac{3}{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)
bad2 1+(0.382+0.923i)T 1 + (0.382 + 0.923i)T
5 1+(−1.79−1.32i)T 1 + (-1.79 - 1.32i)T
17 1+(−1.93−3.64i)T 1 + (-1.93 - 3.64i)T
good3 1+(0.189−0.283i)T+(−1.14−2.77i)T2 1 + (0.189 - 0.283i)T + (-1.14 - 2.77i)T^{2}
7 1+(0.701−3.52i)T+(−6.46−2.67i)T2 1 + (0.701 - 3.52i)T + (-6.46 - 2.67i)T^{2}
11 1+(−0.940+4.72i)T+(−10.1−4.20i)T2 1 + (-0.940 + 4.72i)T + (-10.1 - 4.20i)T^{2}
13 1+1.94iT−13T2 1 + 1.94iT - 13T^{2}
19 1+(−0.750+1.81i)T+(−13.4−13.4i)T2 1 + (-0.750 + 1.81i)T + (-13.4 - 13.4i)T^{2}
23 1+(1.30−0.870i)T+(8.80−21.2i)T2 1 + (1.30 - 0.870i)T + (8.80 - 21.2i)T^{2}
29 1+(−1.38+2.06i)T+(−11.0−26.7i)T2 1 + (-1.38 + 2.06i)T + (-11.0 - 26.7i)T^{2}
31 1+(1.47+7.43i)T+(−28.6+11.8i)T2 1 + (1.47 + 7.43i)T + (-28.6 + 11.8i)T^{2}
37 1+(9.32+6.22i)T+(14.1+34.1i)T2 1 + (9.32 + 6.22i)T + (14.1 + 34.1i)T^{2}
41 1+(−2.81−4.20i)T+(−15.6+37.8i)T2 1 + (-2.81 - 4.20i)T + (-15.6 + 37.8i)T^{2}
43 1+(1.90−4.60i)T+(−30.4−30.4i)T2 1 + (1.90 - 4.60i)T + (-30.4 - 30.4i)T^{2}
47 1+5.22T+47T2 1 + 5.22T + 47T^{2}
53 1+(−6.51+2.69i)T+(37.4−37.4i)T2 1 + (-6.51 + 2.69i)T + (37.4 - 37.4i)T^{2}
59 1+(−0.333+0.138i)T+(41.7−41.7i)T2 1 + (-0.333 + 0.138i)T + (41.7 - 41.7i)T^{2}
61 1+(−4.07+2.72i)T+(23.3−56.3i)T2 1 + (-4.07 + 2.72i)T + (23.3 - 56.3i)T^{2}
67 1+(−8.03+8.03i)T−67iT2 1 + (-8.03 + 8.03i)T - 67iT^{2}
71 1+(−7.79+1.54i)T+(65.5−27.1i)T2 1 + (-7.79 + 1.54i)T + (65.5 - 27.1i)T^{2}
73 1+(3.19+16.0i)T+(−67.4+27.9i)T2 1 + (3.19 + 16.0i)T + (-67.4 + 27.9i)T^{2}
79 1+(16.8+3.34i)T+(72.9+30.2i)T2 1 + (16.8 + 3.34i)T + (72.9 + 30.2i)T^{2}
83 1+(−1.13−2.74i)T+(−58.6+58.6i)T2 1 + (-1.13 - 2.74i)T + (-58.6 + 58.6i)T^{2}
89 1+(−9.67+9.67i)T−89iT2 1 + (-9.67 + 9.67i)T - 89iT^{2}
97 1+(−1.24−6.26i)T+(−89.6+37.1i)T2 1 + (-1.24 - 6.26i)T + (-89.6 + 37.1i)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

−12.83240819620084303402242882063, −11.60144542537565204450958329177, −10.80330109144958828093977448801, −9.916331706657368328753875830600, −8.945435419261279564053446096325, −7.927330657085038154008099201771, −6.16414598537608559351747783608, −5.37746117588323130298315188883, −3.36591525316520460297074542186, −2.14008188539804484405924630620, 1.32818921265181370590130651550, 4.03499802845743405195268873742, 5.20249075679800211366327018708, 6.80541433988191625623914553054, 7.10213463761212312185117257347, 8.744878254311385412246709426798, 9.829176411403294324119453827278, 10.17360930036263711368155906359, 12.01742644059468545479437248069, 12.79900967811055151799907923117

Graph of the ZZ-function along the critical line