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Properties

Label 2-13e2-13.12-c1-0-6
Degree $2$
Conductor $169$
Sign $0.691 + 0.722i$
Analytic cond. $1.34947$
Root an. cond. $1.16166$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.554i·2-s + 0.801·3-s + 1.69·4-s − 2.80i·5-s − 0.445i·6-s + 2.69i·7-s − 2.04i·8-s − 2.35·9-s − 1.55·10-s + 1.19i·11-s + 1.35·12-s + 1.49·14-s − 2.24i·15-s + 2.24·16-s − 1.13·17-s + 1.30i·18-s + ⋯
L(s)  = 1  − 0.392i·2-s + 0.462·3-s + 0.846·4-s − 1.25i·5-s − 0.181i·6-s + 1.01i·7-s − 0.724i·8-s − 0.785·9-s − 0.491·10-s + 0.361i·11-s + 0.391·12-s + 0.399·14-s − 0.580i·15-s + 0.561·16-s − 0.275·17-s + 0.308i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.691 + 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.691 + 0.722i$
Analytic conductor: \(1.34947\)
Root analytic conductor: \(1.16166\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (168, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :1/2),\ 0.691 + 0.722i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35538 - 0.578613i\)
\(L(\frac12)\) \(\approx\) \(1.35538 - 0.578613i\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.554iT - 2T^{2} \)
3 \( 1 - 0.801T + 3T^{2} \)
5 \( 1 + 2.80iT - 5T^{2} \)
7 \( 1 - 2.69iT - 7T^{2} \)
11 \( 1 - 1.19iT - 11T^{2} \)
17 \( 1 + 1.13T + 17T^{2} \)
19 \( 1 - 1.93iT - 19T^{2} \)
23 \( 1 - 4.60T + 23T^{2} \)
29 \( 1 + 7.89T + 29T^{2} \)
31 \( 1 - 5.89iT - 31T^{2} \)
37 \( 1 + 0.951iT - 37T^{2} \)
41 \( 1 - 3.31iT - 41T^{2} \)
43 \( 1 + 7.15T + 43T^{2} \)
47 \( 1 - 7.69iT - 47T^{2} \)
53 \( 1 - 5.87T + 53T^{2} \)
59 \( 1 - 0.0120iT - 59T^{2} \)
61 \( 1 + 8.03T + 61T^{2} \)
67 \( 1 + 9.25iT - 67T^{2} \)
71 \( 1 + 13.7iT - 71T^{2} \)
73 \( 1 + 12.8iT - 73T^{2} \)
79 \( 1 - 0.807T + 79T^{2} \)
83 \( 1 + 16.3iT - 83T^{2} \)
89 \( 1 - 14.7iT - 89T^{2} \)
97 \( 1 - 3.13iT - 97T^{2} \)
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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

−12.46411806192441046530082690304, −11.86302020321111432584732948672, −10.83927107409341649636334428678, −9.375482347468606611056271135866, −8.778823233459227880407182999975, −7.66471625855674818277110366204, −6.12411235041285051012484329717, −5.02991880792359541495104972319, −3.21612061595007245431824799924, −1.85363386397674879286724229876, 2.50507082103092567003728858686, 3.58631967682433352224547082499, 5.67399485917868166993388689587, 6.86912401101177441015696700849, 7.41409192544279426129151621974, 8.614244742138536684828859967988, 10.10481225846461631473521577550, 11.10911071766157646178444943363, 11.42699065598586314950194827702, 13.23172843026177316489265010637

Graph of the $Z$-function along the critical line