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Properties

Label 2-13e2-13.12-c7-0-77
Degree 22
Conductor 169169
Sign −0.277+0.960i-0.277 + 0.960i
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  − 0.679i·2-s + 62.4·3-s + 127.·4-s − 439. i·5-s − 42.4i·6-s − 1.30e3i·7-s − 173. i·8-s + 1.71e3·9-s − 298.·10-s − 574. i·11-s + 7.96e3·12-s − 884.·14-s − 2.74e4i·15-s + 1.62e4·16-s + 1.12e4·17-s − 1.16e3i·18-s + ⋯
L(s)  = 1  − 0.0600i·2-s + 1.33·3-s + 0.996·4-s − 1.57i·5-s − 0.0801i·6-s − 1.43i·7-s − 0.119i·8-s + 0.783·9-s − 0.0943·10-s − 0.130i·11-s + 1.33·12-s − 0.0861·14-s − 2.09i·15-s + 0.989·16-s + 0.556·17-s − 0.0470i·18-s + ⋯

Functional equation

Λ(s)=(169s/2 ΓC(s) L(s)=((−0.277+0.960i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.277 + 0.960i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(169s/2 ΓC(s+7/2) L(s)=((−0.277+0.960i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 169169    =    13213^{2}
Sign: −0.277+0.960i-0.277 + 0.960i
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), −0.277+0.960i)(2,\ 169,\ (\ :7/2),\ -0.277 + 0.960i)

Particular Values

L(4)L(4) ≈\approx 4.3649311684.364931168
L(12)L(\frac12) ≈\approx 4.3649311684.364931168
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+0.679iT−128T2 1 + 0.679iT - 128T^{2}
3 1−62.4T+2.18e3T2 1 - 62.4T + 2.18e3T^{2}
5 1+439.iT−7.81e4T2 1 + 439. iT - 7.81e4T^{2}
7 1+1.30e3iT−8.23e5T2 1 + 1.30e3iT - 8.23e5T^{2}
11 1+574.iT−1.94e7T2 1 + 574. iT - 1.94e7T^{2}
17 1−1.12e4T+4.10e8T2 1 - 1.12e4T + 4.10e8T^{2}
19 1−4.10e4iT−8.93e8T2 1 - 4.10e4iT - 8.93e8T^{2}
23 1−4.42e4T+3.40e9T2 1 - 4.42e4T + 3.40e9T^{2}
29 1+1.45e5T+1.72e10T2 1 + 1.45e5T + 1.72e10T^{2}
31 1+1.22e5iT−2.75e10T2 1 + 1.22e5iT - 2.75e10T^{2}
37 1+4.20e4iT−9.49e10T2 1 + 4.20e4iT - 9.49e10T^{2}
41 1−8.76e4iT−1.94e11T2 1 - 8.76e4iT - 1.94e11T^{2}
43 1−7.49e5T+2.71e11T2 1 - 7.49e5T + 2.71e11T^{2}
47 1−9.40e5iT−5.06e11T2 1 - 9.40e5iT - 5.06e11T^{2}
53 1+9.24e5T+1.17e12T2 1 + 9.24e5T + 1.17e12T^{2}
59 1−6.32e5iT−2.48e12T2 1 - 6.32e5iT - 2.48e12T^{2}
61 1+6.44e4T+3.14e12T2 1 + 6.44e4T + 3.14e12T^{2}
67 1−1.94e6iT−6.06e12T2 1 - 1.94e6iT - 6.06e12T^{2}
71 1+4.72e6iT−9.09e12T2 1 + 4.72e6iT - 9.09e12T^{2}
73 1+1.92e6iT−1.10e13T2 1 + 1.92e6iT - 1.10e13T^{2}
79 1+1.50e6T+1.92e13T2 1 + 1.50e6T + 1.92e13T^{2}
83 1−1.87e6iT−2.71e13T2 1 - 1.87e6iT - 2.71e13T^{2}
89 1+4.37e6iT−4.42e13T2 1 + 4.37e6iT - 4.42e13T^{2}
97 1+1.77e6iT−8.07e13T2 1 + 1.77e6iT - 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

−11.13537051044486418551071057085, −9.993984046130347293686461612051, −9.095060904004543685195907615462, −7.894460781282075737452949829100, −7.53354971571117789724717197621, −5.83358179415957682640647214564, −4.25673496345806576617069962156, −3.32725826919622788498347917240, −1.77569013931811317115581596312, −0.906863534468228296462009163150, 2.04823502054747035473090639917, 2.71764911251562136727618335815, 3.33056681055872930494605730692, 5.61185193615562784602982553979, 6.79434945879940001105142766874, 7.52677736023996231087455014014, 8.683821605932493888748796583076, 9.646384107169439617651997374753, 10.86769415478333527641450731479, 11.58281570568572309631898587626

Graph of the ZZ-function along the critical line