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Properties

Label 2-13e2-169.17-c1-0-8
Degree 22
Conductor 169169
Sign −0.0700+0.997i-0.0700 + 0.997i
Analytic cond. 1.349471.34947
Root an. cond. 1.161661.16166
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  + (−1.99 + 0.407i)2-s + (1.34 + 0.217i)3-s + (1.97 − 0.842i)4-s + (−0.647 − 2.62i)5-s + (−2.76 + 0.111i)6-s + (−3.12 − 1.48i)7-s + (−0.250 + 0.172i)8-s + (−1.09 − 0.365i)9-s + (2.36 + 4.97i)10-s + (−0.457 − 1.36i)11-s + (2.83 − 0.698i)12-s + (−2.59 + 2.50i)13-s + (6.84 + 1.68i)14-s + (−0.295 − 3.65i)15-s + (−2.54 + 2.65i)16-s + (3.18 − 6.71i)17-s + ⋯
L(s)  = 1  + (−1.41 + 0.288i)2-s + (0.773 + 0.125i)3-s + (0.988 − 0.421i)4-s + (−0.289 − 1.17i)5-s + (−1.12 + 0.0454i)6-s + (−1.18 − 0.560i)7-s + (−0.0885 + 0.0611i)8-s + (−0.365 − 0.121i)9-s + (0.746 + 1.57i)10-s + (−0.137 − 0.412i)11-s + (0.818 − 0.201i)12-s + (−0.718 + 0.695i)13-s + (1.82 + 0.450i)14-s + (−0.0762 − 0.944i)15-s + (−0.637 + 0.663i)16-s + (0.772 − 1.62i)17-s + ⋯

Functional equation

Λ(s)=(169s/2 ΓC(s) L(s)=((−0.0700+0.997i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0700 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(169s/2 ΓC(s+1/2) L(s)=((−0.0700+0.997i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0700 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 169169    =    13213^{2}
Sign: −0.0700+0.997i-0.0700 + 0.997i
Analytic conductor: 1.349471.34947
Root analytic conductor: 1.161661.16166
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ169(17,⋅)\chi_{169} (17, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 169, ( :1/2), −0.0700+0.997i)(2,\ 169,\ (\ :1/2),\ -0.0700 + 0.997i)

Particular Values

L(1)L(1) ≈\approx 0.322060−0.345471i0.322060 - 0.345471i
L(12)L(\frac12) ≈\approx 0.322060−0.345471i0.322060 - 0.345471i
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)
bad13 1+(2.59−2.50i)T 1 + (2.59 - 2.50i)T
good2 1+(1.99−0.407i)T+(1.83−0.783i)T2 1 + (1.99 - 0.407i)T + (1.83 - 0.783i)T^{2}
3 1+(−1.34−0.217i)T+(2.84+0.950i)T2 1 + (-1.34 - 0.217i)T + (2.84 + 0.950i)T^{2}
5 1+(0.647+2.62i)T+(−4.42+2.32i)T2 1 + (0.647 + 2.62i)T + (-4.42 + 2.32i)T^{2}
7 1+(3.12+1.48i)T+(4.42+5.42i)T2 1 + (3.12 + 1.48i)T + (4.42 + 5.42i)T^{2}
11 1+(0.457+1.36i)T+(−8.79+6.60i)T2 1 + (0.457 + 1.36i)T + (-8.79 + 6.60i)T^{2}
17 1+(−3.18+6.71i)T+(−10.7−13.1i)T2 1 + (-3.18 + 6.71i)T + (-10.7 - 13.1i)T^{2}
19 1+(−1.35+0.783i)T+(9.5−16.4i)T2 1 + (-1.35 + 0.783i)T + (9.5 - 16.4i)T^{2}
23 1+(−2.33+4.04i)T+(−11.5−19.9i)T2 1 + (-2.33 + 4.04i)T + (-11.5 - 19.9i)T^{2}
29 1+(−1.83−8.97i)T+(−26.6+11.3i)T2 1 + (-1.83 - 8.97i)T + (-26.6 + 11.3i)T^{2}
31 1+(−2.10+4.00i)T+(−17.6−25.5i)T2 1 + (-2.10 + 4.00i)T + (-17.6 - 25.5i)T^{2}
37 1+(−2.23−3.52i)T+(−15.8+33.4i)T2 1 + (-2.23 - 3.52i)T + (-15.8 + 33.4i)T^{2}
41 1+(0.705−4.34i)T+(−38.8−12.9i)T2 1 + (0.705 - 4.34i)T + (-38.8 - 12.9i)T^{2}
43 1+(−6.97−4.41i)T+(18.4+38.8i)T2 1 + (-6.97 - 4.41i)T + (18.4 + 38.8i)T^{2}
47 1+(4.62−0.562i)T+(45.6−11.2i)T2 1 + (4.62 - 0.562i)T + (45.6 - 11.2i)T^{2}
53 1+(6.04+8.76i)T+(−18.7+49.5i)T2 1 + (6.04 + 8.76i)T + (-18.7 + 49.5i)T^{2}
59 1+(−2.94+2.82i)T+(2.37−58.9i)T2 1 + (-2.94 + 2.82i)T + (2.37 - 58.9i)T^{2}
61 1+(1.14+0.0923i)T+(60.2+9.78i)T2 1 + (1.14 + 0.0923i)T + (60.2 + 9.78i)T^{2}
67 1+(−2.26+5.30i)T+(−46.4−48.3i)T2 1 + (-2.26 + 5.30i)T + (-46.4 - 48.3i)T^{2}
71 1+(5.41+4.42i)T+(14.2+69.5i)T2 1 + (5.41 + 4.42i)T + (14.2 + 69.5i)T^{2}
73 1+(−0.379+0.428i)T+(−8.79−72.4i)T2 1 + (-0.379 + 0.428i)T + (-8.79 - 72.4i)T^{2}
79 1+(−0.816−6.72i)T+(−76.7+18.9i)T2 1 + (-0.816 - 6.72i)T + (-76.7 + 18.9i)T^{2}
83 1+(1.28+0.488i)T+(62.1+55.0i)T2 1 + (1.28 + 0.488i)T + (62.1 + 55.0i)T^{2}
89 1+(−13.8−8.00i)T+(44.5+77.0i)T2 1 + (-13.8 - 8.00i)T + (44.5 + 77.0i)T^{2}
97 1+(1.48−0.430i)T+(81.9−51.8i)T2 1 + (1.48 - 0.430i)T + (81.9 - 51.8i)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

−12.54564637276778288305908266707, −11.33505295143201825239331527698, −9.826463161415089185926920665879, −9.397395929177219263137164610295, −8.626685267992241114051849341928, −7.67973331115333934645102539843, −6.65091830931659378667856634685, −4.75690352084653212600529660490, −3.06325715258386003847230237213, −0.61976259606704120967196323977, 2.41985480893920638490651372576, 3.31930093741477234729881203435, 5.92382972322012607896022497665, 7.32603132817230540219633548341, 7.975922344004189746330169300017, 9.085783994313366618720973752885, 9.985262905971005014247047506514, 10.59022378885663659784914102172, 11.81800091953986015157766354021, 12.90698178843920932963708227732

Graph of the ZZ-function along the critical line