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Properties

Label 2-170-85.63-c1-0-4
Degree 22
Conductor 170170
Sign 0.754−0.656i0.754 - 0.656i
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  + (0.923 + 0.382i)2-s + (0.0811 + 0.0161i)3-s + (0.707 + 0.707i)4-s + (1.07 + 1.96i)5-s + (0.0688 + 0.0459i)6-s + (0.143 − 0.214i)7-s + (0.382 + 0.923i)8-s + (−2.76 − 1.14i)9-s + (0.240 + 2.22i)10-s + (0.818 − 1.22i)11-s + (0.0459 + 0.0688i)12-s − 0.161i·13-s + (0.214 − 0.143i)14-s + (0.0553 + 0.176i)15-s + i·16-s + (4.08 − 0.561i)17-s + ⋯
L(s)  = 1  + (0.653 + 0.270i)2-s + (0.0468 + 0.00932i)3-s + (0.353 + 0.353i)4-s + (0.479 + 0.877i)5-s + (0.0280 + 0.0187i)6-s + (0.0541 − 0.0810i)7-s + (0.135 + 0.326i)8-s + (−0.921 − 0.381i)9-s + (0.0759 + 0.703i)10-s + (0.246 − 0.369i)11-s + (0.0132 + 0.0198i)12-s − 0.0448i·13-s + (0.0573 − 0.0382i)14-s + (0.0143 + 0.0456i)15-s + 0.250i·16-s + (0.990 − 0.136i)17-s + ⋯

Functional equation

Λ(s)=(170s/2 ΓC(s) L(s)=((0.754−0.656i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.754 - 0.656i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(170s/2 ΓC(s+1/2) L(s)=((0.754−0.656i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.754 - 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 170170    =    2⋅5⋅172 \cdot 5 \cdot 17
Sign: 0.754−0.656i0.754 - 0.656i
Analytic conductor: 1.357451.35745
Root analytic conductor: 1.165091.16509
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ170(63,⋅)\chi_{170} (63, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 170, ( :1/2), 0.754−0.656i)(2,\ 170,\ (\ :1/2),\ 0.754 - 0.656i)

Particular Values

L(1)L(1) ≈\approx 1.57039+0.588109i1.57039 + 0.588109i
L(12)L(\frac12) ≈\approx 1.57039+0.588109i1.57039 + 0.588109i
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+(−0.923−0.382i)T 1 + (-0.923 - 0.382i)T
5 1+(−1.07−1.96i)T 1 + (-1.07 - 1.96i)T
17 1+(−4.08+0.561i)T 1 + (-4.08 + 0.561i)T
good3 1+(−0.0811−0.0161i)T+(2.77+1.14i)T2 1 + (-0.0811 - 0.0161i)T + (2.77 + 1.14i)T^{2}
7 1+(−0.143+0.214i)T+(−2.67−6.46i)T2 1 + (-0.143 + 0.214i)T + (-2.67 - 6.46i)T^{2}
11 1+(−0.818+1.22i)T+(−4.20−10.1i)T2 1 + (-0.818 + 1.22i)T + (-4.20 - 10.1i)T^{2}
13 1+0.161iT−13T2 1 + 0.161iT - 13T^{2}
19 1+(1.05−0.436i)T+(13.4−13.4i)T2 1 + (1.05 - 0.436i)T + (13.4 - 13.4i)T^{2}
23 1+(1.45+7.32i)T+(−21.2+8.80i)T2 1 + (1.45 + 7.32i)T + (-21.2 + 8.80i)T^{2}
29 1+(5.25+1.04i)T+(26.7+11.0i)T2 1 + (5.25 + 1.04i)T + (26.7 + 11.0i)T^{2}
31 1+(4.63+6.94i)T+(−11.8+28.6i)T2 1 + (4.63 + 6.94i)T + (-11.8 + 28.6i)T^{2}
37 1+(1.08−5.47i)T+(−34.1−14.1i)T2 1 + (1.08 - 5.47i)T + (-34.1 - 14.1i)T^{2}
41 1+(5.60−1.11i)T+(37.8−15.6i)T2 1 + (5.60 - 1.11i)T + (37.8 - 15.6i)T^{2}
43 1+(−1.92+0.798i)T+(30.4−30.4i)T2 1 + (-1.92 + 0.798i)T + (30.4 - 30.4i)T^{2}
47 1−9.14T+47T2 1 - 9.14T + 47T^{2}
53 1+(−0.350+0.845i)T+(−37.4−37.4i)T2 1 + (-0.350 + 0.845i)T + (-37.4 - 37.4i)T^{2}
59 1+(−0.837+2.02i)T+(−41.7−41.7i)T2 1 + (-0.837 + 2.02i)T + (-41.7 - 41.7i)T^{2}
61 1+(−0.778−3.91i)T+(−56.3+23.3i)T2 1 + (-0.778 - 3.91i)T + (-56.3 + 23.3i)T^{2}
67 1+(−8.09−8.09i)T+67iT2 1 + (-8.09 - 8.09i)T + 67iT^{2}
71 1+(8.03−5.36i)T+(27.1−65.5i)T2 1 + (8.03 - 5.36i)T + (27.1 - 65.5i)T^{2}
73 1+(−1.72−2.58i)T+(−27.9+67.4i)T2 1 + (-1.72 - 2.58i)T + (-27.9 + 67.4i)T^{2}
79 1+(0.595+0.398i)T+(30.2+72.9i)T2 1 + (0.595 + 0.398i)T + (30.2 + 72.9i)T^{2}
83 1+(−2.66−1.10i)T+(58.6+58.6i)T2 1 + (-2.66 - 1.10i)T + (58.6 + 58.6i)T^{2}
89 1+(9.97+9.97i)T+89iT2 1 + (9.97 + 9.97i)T + 89iT^{2}
97 1+(1.95+2.93i)T+(−37.1+89.6i)T2 1 + (1.95 + 2.93i)T + (-37.1 + 89.6i)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

−13.03398561125957398890096673319, −11.89267578175362794338725952946, −11.07091632680718191277091425365, −10.01207075477229272403439262529, −8.719995640178670499532041978174, −7.49730741372512207875660219250, −6.30694149435017943742716419121, −5.58179527268828560091948486381, −3.80568486617153211652679354691, −2.59821780960171449969443868224, 1.86049610085552615679872933456, 3.61563774823300741128883955328, 5.17220309507120005612337045854, 5.78253885460971068363643655080, 7.41675749926132543230900173441, 8.708563918699751308042904124857, 9.650861939133060357279267136556, 10.82976966977332231385188914061, 11.91031060327337250396807268261, 12.61935039298411868153637265897

Graph of the ZZ-function along the critical line