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Properties

Label 2-13e2-169.127-c1-0-6
Degree 22
Conductor 169169
Sign 0.530−0.847i0.530 - 0.847i
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  + (0.596 − 0.0240i)2-s + (0.703 + 2.43i)3-s + (−1.63 + 0.132i)4-s + (3.70 − 1.40i)5-s + (0.478 + 1.43i)6-s + (−0.324 + 0.762i)7-s + (−2.16 + 0.262i)8-s + (−2.87 + 1.81i)9-s + (2.17 − 0.926i)10-s + (−0.896 + 1.41i)11-s + (−1.47 − 3.88i)12-s + (−2.52 − 2.57i)13-s + (−0.175 + 0.462i)14-s + (6.01 + 8.00i)15-s + (1.96 − 0.318i)16-s + (6.43 + 2.74i)17-s + ⋯
L(s)  = 1  + (0.422 − 0.0170i)2-s + (0.406 + 1.40i)3-s + (−0.818 + 0.0661i)4-s + (1.65 − 0.627i)5-s + (0.195 + 0.585i)6-s + (−0.122 + 0.288i)7-s + (−0.763 + 0.0927i)8-s + (−0.958 + 0.605i)9-s + (0.687 − 0.292i)10-s + (−0.270 + 0.427i)11-s + (−0.425 − 1.12i)12-s + (−0.699 − 0.714i)13-s + (−0.0469 + 0.123i)14-s + (1.55 + 2.06i)15-s + (0.490 − 0.0796i)16-s + (1.56 + 0.665i)17-s + ⋯

Functional equation

Λ(s)=(169s/2 ΓC(s) L(s)=((0.530−0.847i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.530 - 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(169s/2 ΓC(s+1/2) L(s)=((0.530−0.847i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.530 - 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 169169    =    13213^{2}
Sign: 0.530−0.847i0.530 - 0.847i
Analytic conductor: 1.349471.34947
Root analytic conductor: 1.161661.16166
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ169(127,⋅)\chi_{169} (127, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 169, ( :1/2), 0.530−0.847i)(2,\ 169,\ (\ :1/2),\ 0.530 - 0.847i)

Particular Values

L(1)L(1) ≈\approx 1.35928+0.752856i1.35928 + 0.752856i
L(12)L(\frac12) ≈\approx 1.35928+0.752856i1.35928 + 0.752856i
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+(2.52+2.57i)T 1 + (2.52 + 2.57i)T
good2 1+(−0.596+0.0240i)T+(1.99−0.160i)T2 1 + (-0.596 + 0.0240i)T + (1.99 - 0.160i)T^{2}
3 1+(−0.703−2.43i)T+(−2.53+1.60i)T2 1 + (-0.703 - 2.43i)T + (-2.53 + 1.60i)T^{2}
5 1+(−3.70+1.40i)T+(3.74−3.31i)T2 1 + (-3.70 + 1.40i)T + (3.74 - 3.31i)T^{2}
7 1+(0.324−0.762i)T+(−4.84−5.04i)T2 1 + (0.324 - 0.762i)T + (-4.84 - 5.04i)T^{2}
11 1+(0.896−1.41i)T+(−4.71−9.93i)T2 1 + (0.896 - 1.41i)T + (-4.71 - 9.93i)T^{2}
17 1+(−6.43−2.74i)T+(11.7+12.2i)T2 1 + (-6.43 - 2.74i)T + (11.7 + 12.2i)T^{2}
19 1+(6.17+3.56i)T+(9.5+16.4i)T2 1 + (6.17 + 3.56i)T + (9.5 + 16.4i)T^{2}
23 1+(1.23+2.13i)T+(−11.5+19.9i)T2 1 + (1.23 + 2.13i)T + (-11.5 + 19.9i)T^{2}
29 1+(0.333+8.28i)T+(−28.9+2.33i)T2 1 + (0.333 + 8.28i)T + (-28.9 + 2.33i)T^{2}
31 1+(−0.422+0.476i)T+(−3.73−30.7i)T2 1 + (-0.422 + 0.476i)T + (-3.73 - 30.7i)T^{2}
37 1+(4.24+0.865i)T+(34.0+14.5i)T2 1 + (4.24 + 0.865i)T + (34.0 + 14.5i)T^{2}
41 1+(3.37−0.977i)T+(34.6−21.9i)T2 1 + (3.37 - 0.977i)T + (34.6 - 21.9i)T^{2}
43 1+(−0.218−1.07i)T+(−39.5+16.8i)T2 1 + (-0.218 - 1.07i)T + (-39.5 + 16.8i)T^{2}
47 1+(1.62+1.11i)T+(16.6+43.9i)T2 1 + (1.62 + 1.11i)T + (16.6 + 43.9i)T^{2}
53 1+(0.146+1.21i)T+(−51.4+12.6i)T2 1 + (0.146 + 1.21i)T + (-51.4 + 12.6i)T^{2}
59 1+(1.82−11.2i)T+(−55.9−18.6i)T2 1 + (1.82 - 11.2i)T + (-55.9 - 18.6i)T^{2}
61 1+(1.57+1.18i)T+(16.9+58.5i)T2 1 + (1.57 + 1.18i)T + (16.9 + 58.5i)T^{2}
67 1+(0.248−3.07i)T+(−66.1−10.7i)T2 1 + (0.248 - 3.07i)T + (-66.1 - 10.7i)T^{2}
71 1+(2.34+2.24i)T+(2.85+70.9i)T2 1 + (2.34 + 2.24i)T + (2.85 + 70.9i)T^{2}
73 1+(−4.05−7.71i)T+(−41.4+60.0i)T2 1 + (-4.05 - 7.71i)T + (-41.4 + 60.0i)T^{2}
79 1+(3.06−4.44i)T+(−28.0−73.8i)T2 1 + (3.06 - 4.44i)T + (-28.0 - 73.8i)T^{2}
83 1+(1.84+7.49i)T+(−73.4+38.5i)T2 1 + (1.84 + 7.49i)T + (-73.4 + 38.5i)T^{2}
89 1+(14.6−8.44i)T+(44.5−77.0i)T2 1 + (14.6 - 8.44i)T + (44.5 - 77.0i)T^{2}
97 1+(0.0230−0.0188i)T+(19.4−95.0i)T2 1 + (0.0230 - 0.0188i)T + (19.4 - 95.0i)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

−13.01014485169274814415911187934, −12.38844250300988720927870545413, −10.26657428696973952293999719403, −9.998588179703306840586682935554, −9.144921965515437066481701437403, −8.320537498078993781348396825121, −5.93080903854298529010410017269, −5.17526434169166277470411324840, −4.26945292368217674609900172383, −2.67282562755406300532057182839, 1.74628867828756993236492666591, 3.16778407750757940894655670948, 5.28452575154974425133305177007, 6.24357055998728091756393597880, 7.18781998446995535484314650179, 8.498968384885938443360378988380, 9.603431327959235573532882140094, 10.41509626862670720307827880588, 12.19707425058941366029858197294, 12.85763581531037432616539492165

Graph of the ZZ-function along the critical line