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Properties

Label 2-13e2-13.12-c7-0-1
Degree 22
Conductor 169169
Sign −0.691+0.722i-0.691 + 0.722i
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  + 6.93i·2-s − 64.5·3-s + 79.8·4-s + 33.5i·5-s − 447. i·6-s + 160. i·7-s + 1.44e3i·8-s + 1.97e3·9-s − 232.·10-s − 5.93e3i·11-s − 5.15e3·12-s − 1.11e3·14-s − 2.16e3i·15-s + 216.·16-s − 1.10e4·17-s + 1.37e4i·18-s + ⋯
L(s)  = 1  + 0.613i·2-s − 1.37·3-s + 0.623·4-s + 0.119i·5-s − 0.846i·6-s + 0.177i·7-s + 0.995i·8-s + 0.903·9-s − 0.0735·10-s − 1.34i·11-s − 0.860·12-s − 0.108·14-s − 0.165i·15-s + 0.0131·16-s − 0.545·17-s + 0.553i·18-s + ⋯

Functional equation

Λ(s)=(169s/2 ΓC(s) L(s)=((−0.691+0.722i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(169s/2 ΓC(s+7/2) L(s)=((−0.691+0.722i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 169169    =    13213^{2}
Sign: −0.691+0.722i-0.691 + 0.722i
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), −0.691+0.722i)(2,\ 169,\ (\ :7/2),\ -0.691 + 0.722i)

Particular Values

L(4)L(4) ≈\approx 0.19512086120.1951208612
L(12)L(\frac12) ≈\approx 0.19512086120.1951208612
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 1−6.93iT−128T2 1 - 6.93iT - 128T^{2}
3 1+64.5T+2.18e3T2 1 + 64.5T + 2.18e3T^{2}
5 1−33.5iT−7.81e4T2 1 - 33.5iT - 7.81e4T^{2}
7 1−160.iT−8.23e5T2 1 - 160. iT - 8.23e5T^{2}
11 1+5.93e3iT−1.94e7T2 1 + 5.93e3iT - 1.94e7T^{2}
17 1+1.10e4T+4.10e8T2 1 + 1.10e4T + 4.10e8T^{2}
19 1−5.32e4iT−8.93e8T2 1 - 5.32e4iT - 8.93e8T^{2}
23 1−6.21e4T+3.40e9T2 1 - 6.21e4T + 3.40e9T^{2}
29 1+2.48e5T+1.72e10T2 1 + 2.48e5T + 1.72e10T^{2}
31 1−1.36e5iT−2.75e10T2 1 - 1.36e5iT - 2.75e10T^{2}
37 1+1.84e5iT−9.49e10T2 1 + 1.84e5iT - 9.49e10T^{2}
41 1−2.03e5iT−1.94e11T2 1 - 2.03e5iT - 1.94e11T^{2}
43 1+5.32e5T+2.71e11T2 1 + 5.32e5T + 2.71e11T^{2}
47 1−2.87e5iT−5.06e11T2 1 - 2.87e5iT - 5.06e11T^{2}
53 1+1.37e6T+1.17e12T2 1 + 1.37e6T + 1.17e12T^{2}
59 1+5.14e5iT−2.48e12T2 1 + 5.14e5iT - 2.48e12T^{2}
61 1+1.27e6T+3.14e12T2 1 + 1.27e6T + 3.14e12T^{2}
67 1+4.00e6iT−6.06e12T2 1 + 4.00e6iT - 6.06e12T^{2}
71 1+1.78e6iT−9.09e12T2 1 + 1.78e6iT - 9.09e12T^{2}
73 1+3.26e6iT−1.10e13T2 1 + 3.26e6iT - 1.10e13T^{2}
79 1+6.10e6T+1.92e13T2 1 + 6.10e6T + 1.92e13T^{2}
83 1+4.96e6iT−2.71e13T2 1 + 4.96e6iT - 2.71e13T^{2}
89 1+5.03e6iT−4.42e13T2 1 + 5.03e6iT - 4.42e13T^{2}
97 1−1.33e7iT−8.07e13T2 1 - 1.33e7iT - 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

−11.87939607238252503622461995002, −11.04267227696543718505712756525, −10.62628811389660892975840682098, −8.903574763373508045786336623847, −7.72065707097927129113549168547, −6.55255000499229356967152085109, −5.90743760282369114800761857865, −5.10757929805533291374538367010, −3.25205035311750106171456469274, −1.47071002758832592381223822614, 0.06306322134637245900103755383, 1.26881946581849785027849068562, 2.60369400992848539873172163138, 4.34463490518336322004174914691, 5.34443676119541928213572897449, 6.76360817079861157810504545463, 7.16509314235980259486323899098, 9.171773525521972674729425743566, 10.22308760910262943936908229892, 11.15866678959411019007119256335

Graph of the ZZ-function along the critical line