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Properties

Label 2-13e2-13.2-c2-0-7
Degree 22
Conductor 169169
Sign 0.623+0.781i0.623 + 0.781i
Analytic cond. 4.604914.60491
Root an. cond. 2.145902.14590
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.11 − 0.834i)2-s + (1.02 + 1.77i)3-s + (5.54 + 3.19i)4-s + (5.50 − 5.50i)5-s + (−1.70 − 6.37i)6-s + (−3.16 + 0.846i)7-s + (−5.47 − 5.47i)8-s + (2.40 − 4.16i)9-s + (−21.7 + 12.5i)10-s + (−1.31 + 4.90i)11-s + 13.0i·12-s + 10.5·14-s + (15.3 + 4.12i)15-s + (−0.322 − 0.558i)16-s + (8.72 + 5.03i)17-s + (−10.9 + 10.9i)18-s + ⋯
L(s)  = 1  + (−1.55 − 0.417i)2-s + (0.341 + 0.590i)3-s + (1.38 + 0.799i)4-s + (1.10 − 1.10i)5-s + (−0.284 − 1.06i)6-s + (−0.451 + 0.120i)7-s + (−0.683 − 0.683i)8-s + (0.267 − 0.463i)9-s + (−2.17 + 1.25i)10-s + (−0.119 + 0.445i)11-s + 1.09i·12-s + 0.753·14-s + (1.02 + 0.274i)15-s + (−0.0201 − 0.0349i)16-s + (0.513 + 0.296i)17-s + (−0.609 + 0.609i)18-s + ⋯

Functional equation

Λ(s)=(169s/2 ΓC(s) L(s)=((0.623+0.781i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.623 + 0.781i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(169s/2 ΓC(s+1) L(s)=((0.623+0.781i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.623 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 169169    =    13213^{2}
Sign: 0.623+0.781i0.623 + 0.781i
Analytic conductor: 4.604914.60491
Root analytic conductor: 2.145902.14590
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ169(80,⋅)\chi_{169} (80, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 169, ( :1), 0.623+0.781i)(2,\ 169,\ (\ :1),\ 0.623 + 0.781i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.823796−0.396763i0.823796 - 0.396763i
L(12)L(\frac12) ≈\approx 0.823796−0.396763i0.823796 - 0.396763i
L(2)L(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 1
good2 1+(3.11+0.834i)T+(3.46+2i)T2 1 + (3.11 + 0.834i)T + (3.46 + 2i)T^{2}
3 1+(−1.02−1.77i)T+(−4.5+7.79i)T2 1 + (-1.02 - 1.77i)T + (-4.5 + 7.79i)T^{2}
5 1+(−5.50+5.50i)T−25iT2 1 + (-5.50 + 5.50i)T - 25iT^{2}
7 1+(3.16−0.846i)T+(42.4−24.5i)T2 1 + (3.16 - 0.846i)T + (42.4 - 24.5i)T^{2}
11 1+(1.31−4.90i)T+(−104.−60.5i)T2 1 + (1.31 - 4.90i)T + (-104. - 60.5i)T^{2}
17 1+(−8.72−5.03i)T+(144.5+250.i)T2 1 + (-8.72 - 5.03i)T + (144.5 + 250. i)T^{2}
19 1+(3.07+11.4i)T+(−312.+180.5i)T2 1 + (3.07 + 11.4i)T + (-312. + 180.5i)T^{2}
23 1+(−27.5+15.9i)T+(264.5−458.i)T2 1 + (-27.5 + 15.9i)T + (264.5 - 458. i)T^{2}
29 1+(16.3+28.2i)T+(−420.5+728.i)T2 1 + (16.3 + 28.2i)T + (-420.5 + 728. i)T^{2}
31 1+(−8.05+8.05i)T−961iT2 1 + (-8.05 + 8.05i)T - 961iT^{2}
37 1+(−11.4+42.7i)T+(−1.18e3−684.5i)T2 1 + (-11.4 + 42.7i)T + (-1.18e3 - 684.5i)T^{2}
41 1+(−45.6−12.2i)T+(1.45e3+840.5i)T2 1 + (-45.6 - 12.2i)T + (1.45e3 + 840.5i)T^{2}
43 1+(−20.1−11.6i)T+(924.5+1.60e3i)T2 1 + (-20.1 - 11.6i)T + (924.5 + 1.60e3i)T^{2}
47 1+(−64.6−64.6i)T+2.20e3iT2 1 + (-64.6 - 64.6i)T + 2.20e3iT^{2}
53 1+48.7T+2.80e3T2 1 + 48.7T + 2.80e3T^{2}
59 1+(23.5−6.31i)T+(3.01e3−1.74e3i)T2 1 + (23.5 - 6.31i)T + (3.01e3 - 1.74e3i)T^{2}
61 1+(6.30−10.9i)T+(−1.86e3−3.22e3i)T2 1 + (6.30 - 10.9i)T + (-1.86e3 - 3.22e3i)T^{2}
67 1+(−16.7−4.48i)T+(3.88e3+2.24e3i)T2 1 + (-16.7 - 4.48i)T + (3.88e3 + 2.24e3i)T^{2}
71 1+(10.4+38.8i)T+(−4.36e3+2.52e3i)T2 1 + (10.4 + 38.8i)T + (-4.36e3 + 2.52e3i)T^{2}
73 1+(76.5+76.5i)T+5.32e3iT2 1 + (76.5 + 76.5i)T + 5.32e3iT^{2}
79 1+94.0T+6.24e3T2 1 + 94.0T + 6.24e3T^{2}
83 1+(33.6−33.6i)T−6.88e3iT2 1 + (33.6 - 33.6i)T - 6.88e3iT^{2}
89 1+(9.39−35.0i)T+(−6.85e3−3.96e3i)T2 1 + (9.39 - 35.0i)T + (-6.85e3 - 3.96e3i)T^{2}
97 1+(−10.7−40.1i)T+(−8.14e3+4.70e3i)T2 1 + (-10.7 - 40.1i)T + (-8.14e3 + 4.70e3i)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

−12.39893746351157645278052581707, −10.92997551735630090469630904665, −9.970114589987452088552911722058, −9.263098967936142922475680428269, −8.998022855513645814887861141494, −7.62457284787183206338454480440, −6.14296158577976590342652459362, −4.57935272577756933731728372391, −2.57656874567734805831315343369, −1.01445487564334024486990373070, 1.51557483897666923047778109158, 2.89347416876358866538589217714, 5.75372370587984802745357989225, 6.86699348123274564144303927029, 7.41181466711533650816275581904, 8.613028364233845104979939442840, 9.675886173425645541891415449394, 10.35410080224383763300898454572, 11.12915140852926141118624692469, 12.88715852891347977150662047081

Graph of the ZZ-function along the critical line