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Properties

Label 2-13e2-13.12-c7-0-69
Degree 22
Conductor 169169
Sign −0.832+0.554i-0.832 + 0.554i
Analytic cond. 52.793052.7930
Root an. cond. 7.265887.26588
Motivic weight 77
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6.08i·2-s + 25.6·3-s + 90.9·4-s + 442. i·5-s − 155. i·6-s − 761. i·7-s − 1.33e3i·8-s − 1.53e3·9-s + 2.69e3·10-s − 6.11e3i·11-s + 2.33e3·12-s − 4.63e3·14-s + 1.13e4i·15-s + 3.54e3·16-s − 3.75e4·17-s + 9.30e3i·18-s + ⋯
L(s)  = 1  − 0.537i·2-s + 0.548·3-s + 0.710·4-s + 1.58i·5-s − 0.294i·6-s − 0.839i·7-s − 0.919i·8-s − 0.699·9-s + 0.850·10-s − 1.38i·11-s + 0.389·12-s − 0.451·14-s + 0.867i·15-s + 0.216·16-s − 1.85·17-s + 0.376i·18-s + ⋯

Functional equation

Λ(s)=(169s/2 ΓC(s) L(s)=((−0.832+0.554i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(169s/2 ΓC(s+7/2) L(s)=((−0.832+0.554i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 169169    =    13213^{2}
Sign: −0.832+0.554i-0.832 + 0.554i
Analytic conductor: 52.793052.7930
Root analytic conductor: 7.265887.26588
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: χ169(168,⋅)\chi_{169} (168, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 169, ( :7/2), −0.832+0.554i)(2,\ 169,\ (\ :7/2),\ -0.832 + 0.554i)

Particular Values

L(4)L(4) ≈\approx 1.5077312041.507731204
L(12)L(\frac12) ≈\approx 1.5077312041.507731204
L(92)L(\frac{9}{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+6.08iT−128T2 1 + 6.08iT - 128T^{2}
3 1−25.6T+2.18e3T2 1 - 25.6T + 2.18e3T^{2}
5 1−442.iT−7.81e4T2 1 - 442. iT - 7.81e4T^{2}
7 1+761.iT−8.23e5T2 1 + 761. iT - 8.23e5T^{2}
11 1+6.11e3iT−1.94e7T2 1 + 6.11e3iT - 1.94e7T^{2}
17 1+3.75e4T+4.10e8T2 1 + 3.75e4T + 4.10e8T^{2}
19 1−3.39e3iT−8.93e8T2 1 - 3.39e3iT - 8.93e8T^{2}
23 1−2.98e4T+3.40e9T2 1 - 2.98e4T + 3.40e9T^{2}
29 1+4.22e4T+1.72e10T2 1 + 4.22e4T + 1.72e10T^{2}
31 1+1.24e5iT−2.75e10T2 1 + 1.24e5iT - 2.75e10T^{2}
37 1+1.42e5iT−9.49e10T2 1 + 1.42e5iT - 9.49e10T^{2}
41 1+7.14e4iT−1.94e11T2 1 + 7.14e4iT - 1.94e11T^{2}
43 1−1.27e4T+2.71e11T2 1 - 1.27e4T + 2.71e11T^{2}
47 1+4.37e5iT−5.06e11T2 1 + 4.37e5iT - 5.06e11T^{2}
53 1+1.01e6T+1.17e12T2 1 + 1.01e6T + 1.17e12T^{2}
59 1+1.75e6iT−2.48e12T2 1 + 1.75e6iT - 2.48e12T^{2}
61 1+1.69e6T+3.14e12T2 1 + 1.69e6T + 3.14e12T^{2}
67 1−3.41e6iT−6.06e12T2 1 - 3.41e6iT - 6.06e12T^{2}
71 1+7.94e5iT−9.09e12T2 1 + 7.94e5iT - 9.09e12T^{2}
73 1+3.45e6iT−1.10e13T2 1 + 3.45e6iT - 1.10e13T^{2}
79 1−6.86e6T+1.92e13T2 1 - 6.86e6T + 1.92e13T^{2}
83 1+8.04e6iT−2.71e13T2 1 + 8.04e6iT - 2.71e13T^{2}
89 1−1.04e6iT−4.42e13T2 1 - 1.04e6iT - 4.42e13T^{2}
97 1−1.12e6iT−8.07e13T2 1 - 1.12e6iT - 8.07e13T^{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

−11.01883215499156510589234734589, −10.53470947382804893407822784745, −9.151227873606653693468743795214, −7.83017300376142524617741639311, −6.83837184544700636474125966248, −6.08764458354798407934828102684, −3.78521356703745734755941469579, −3.02392806469875763319021746088, −2.15963351939169272562687605739, −0.31046053490682974188110448157, 1.68332094634570567120522687748, 2.58491151263156800527334546718, 4.56286045338036693677559561532, 5.43402158103082561362038619090, 6.66781976316938354468471022592, 7.933958511188083648795183485527, 8.787637668857675938594935898361, 9.344931979027043292945031046061, 11.05466799468092322355906249018, 12.06889390572161552111644043656

Graph of the ZZ-function along the critical line