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Properties

Label 2-13e2-13.12-c7-0-80
Degree 22
Conductor 169169
Sign 0.999+0.0304i0.999 + 0.0304i
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  − 15.0i·2-s − 85.3·3-s − 97.6·4-s − 439. i·5-s + 1.28e3i·6-s − 536. i·7-s − 455. i·8-s + 5.10e3·9-s − 6.60e3·10-s − 2.99e3i·11-s + 8.34e3·12-s − 8.06e3·14-s + 3.75e4i·15-s − 1.93e4·16-s − 2.23e4·17-s − 7.66e4i·18-s + ⋯
L(s)  = 1  − 1.32i·2-s − 1.82·3-s − 0.763·4-s − 1.57i·5-s + 2.42i·6-s − 0.591i·7-s − 0.314i·8-s + 2.33·9-s − 2.08·10-s − 0.678i·11-s + 1.39·12-s − 0.785·14-s + 2.87i·15-s − 1.18·16-s − 1.10·17-s − 3.09i·18-s + ⋯

Functional equation

Λ(s)=(169s/2 ΓC(s) L(s)=((0.999+0.0304i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0304i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(169s/2 ΓC(s+7/2) L(s)=((0.999+0.0304i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0304i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 169169    =    13213^{2}
Sign: 0.999+0.0304i0.999 + 0.0304i
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), 0.999+0.0304i)(2,\ 169,\ (\ :7/2),\ 0.999 + 0.0304i)

Particular Values

L(4)L(4) ≈\approx 0.38589144970.3858914497
L(12)L(\frac12) ≈\approx 0.38589144970.3858914497
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+15.0iT−128T2 1 + 15.0iT - 128T^{2}
3 1+85.3T+2.18e3T2 1 + 85.3T + 2.18e3T^{2}
5 1+439.iT−7.81e4T2 1 + 439. iT - 7.81e4T^{2}
7 1+536.iT−8.23e5T2 1 + 536. iT - 8.23e5T^{2}
11 1+2.99e3iT−1.94e7T2 1 + 2.99e3iT - 1.94e7T^{2}
17 1+2.23e4T+4.10e8T2 1 + 2.23e4T + 4.10e8T^{2}
19 1+1.80e4iT−8.93e8T2 1 + 1.80e4iT - 8.93e8T^{2}
23 1+3.57e4T+3.40e9T2 1 + 3.57e4T + 3.40e9T^{2}
29 1+1.13e5T+1.72e10T2 1 + 1.13e5T + 1.72e10T^{2}
31 1−1.65e5iT−2.75e10T2 1 - 1.65e5iT - 2.75e10T^{2}
37 1+4.40e5iT−9.49e10T2 1 + 4.40e5iT - 9.49e10T^{2}
41 1−1.06e5iT−1.94e11T2 1 - 1.06e5iT - 1.94e11T^{2}
43 1−5.08e5T+2.71e11T2 1 - 5.08e5T + 2.71e11T^{2}
47 1+3.23e5iT−5.06e11T2 1 + 3.23e5iT - 5.06e11T^{2}
53 1+1.34e6T+1.17e12T2 1 + 1.34e6T + 1.17e12T^{2}
59 1+1.16e6iT−2.48e12T2 1 + 1.16e6iT - 2.48e12T^{2}
61 1−1.37e6T+3.14e12T2 1 - 1.37e6T + 3.14e12T^{2}
67 1+1.02e6iT−6.06e12T2 1 + 1.02e6iT - 6.06e12T^{2}
71 1−4.68e6iT−9.09e12T2 1 - 4.68e6iT - 9.09e12T^{2}
73 1−4.77e5iT−1.10e13T2 1 - 4.77e5iT - 1.10e13T^{2}
79 1+6.05e5T+1.92e13T2 1 + 6.05e5T + 1.92e13T^{2}
83 1+5.20e6iT−2.71e13T2 1 + 5.20e6iT - 2.71e13T^{2}
89 1+1.17e7iT−4.42e13T2 1 + 1.17e7iT - 4.42e13T^{2}
97 1+3.49e6iT−8.07e13T2 1 + 3.49e6iT - 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

−10.94277640282271663087547637869, −9.852180863242144516227465331504, −8.845927545162082608670446003578, −7.08567197430881459887649379923, −5.82424203843503146983173569017, −4.74836820983702235870767300718, −4.00830495919832284476172376660, −1.73285787175176946316253414307, −0.71017144402759382569959113449, −0.19797295310384042410238860145, 2.16007091279279199839598599737, 4.33049543879672183909121777143, 5.57847306042422706404608565993, 6.28472283158591525166154744786, 6.87989611399538890827773523328, 7.76272550402683444289883621004, 9.605031092237755392709386158289, 10.71776518989388986335539119898, 11.34912218504964582832214189494, 12.22585326721736472723024652982

Graph of the ZZ-function along the critical line