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Properties

Label 2-13e2-13.12-c7-0-80
Degree $2$
Conductor $169$
Sign $0.999 + 0.0304i$
Analytic cond. $52.7930$
Root an. cond. $7.26588$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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

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

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.999 + 0.0304i$
Analytic conductor: \(52.7930\)
Root analytic conductor: \(7.26588\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (168, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :7/2),\ 0.999 + 0.0304i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.3858914497\)
\(L(\frac12)\) \(\approx\) \(0.3858914497\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
good2 \( 1 + 15.0iT - 128T^{2} \)
3 \( 1 + 85.3T + 2.18e3T^{2} \)
5 \( 1 + 439. iT - 7.81e4T^{2} \)
7 \( 1 + 536. iT - 8.23e5T^{2} \)
11 \( 1 + 2.99e3iT - 1.94e7T^{2} \)
17 \( 1 + 2.23e4T + 4.10e8T^{2} \)
19 \( 1 + 1.80e4iT - 8.93e8T^{2} \)
23 \( 1 + 3.57e4T + 3.40e9T^{2} \)
29 \( 1 + 1.13e5T + 1.72e10T^{2} \)
31 \( 1 - 1.65e5iT - 2.75e10T^{2} \)
37 \( 1 + 4.40e5iT - 9.49e10T^{2} \)
41 \( 1 - 1.06e5iT - 1.94e11T^{2} \)
43 \( 1 - 5.08e5T + 2.71e11T^{2} \)
47 \( 1 + 3.23e5iT - 5.06e11T^{2} \)
53 \( 1 + 1.34e6T + 1.17e12T^{2} \)
59 \( 1 + 1.16e6iT - 2.48e12T^{2} \)
61 \( 1 - 1.37e6T + 3.14e12T^{2} \)
67 \( 1 + 1.02e6iT - 6.06e12T^{2} \)
71 \( 1 - 4.68e6iT - 9.09e12T^{2} \)
73 \( 1 - 4.77e5iT - 1.10e13T^{2} \)
79 \( 1 + 6.05e5T + 1.92e13T^{2} \)
83 \( 1 + 5.20e6iT - 2.71e13T^{2} \)
89 \( 1 + 1.17e7iT - 4.42e13T^{2} \)
97 \( 1 + 3.49e6iT - 8.07e13T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

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 $Z$-function along the critical line