↑

Properties

Label 2-13e2-13.12-c7-0-23
Degree $2$
Conductor $169$
Sign $-0.832 + 0.554i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 6.78i·2-s − 30.1·3-s + 81.9·4-s + 93.2i·5-s − 204. i·6-s + 1.61e3i·7-s + 1.42e3i·8-s − 1.27e3·9-s − 632.·10-s + 6.35e3i·11-s − 2.47e3·12-s − 1.09e4·14-s − 2.81e3i·15-s + 821.·16-s + 8.37e3·17-s − 8.66e3i·18-s + ⋯
L(s)  = 1  + 0.599i·2-s − 0.645·3-s + 0.640·4-s + 0.333i·5-s − 0.387i·6-s + 1.78i·7-s + 0.983i·8-s − 0.583·9-s − 0.200·10-s + 1.44i·11-s − 0.413·12-s − 1.06·14-s − 0.215i·15-s + 0.0501·16-s + 0.413·17-s − 0.350i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $-0.832 + 0.554i$
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: \((2,\ 169,\ (\ :7/2),\ -0.832 + 0.554i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.565099129\)
\(L(\frac12)\) \(\approx\) \(1.565099129\)
\(L(\frac{9}{2})\) 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 \( 1 - 6.78iT - 128T^{2} \)
3 \( 1 + 30.1T + 2.18e3T^{2} \)
5 \( 1 - 93.2iT - 7.81e4T^{2} \)
7 \( 1 - 1.61e3iT - 8.23e5T^{2} \)
11 \( 1 - 6.35e3iT - 1.94e7T^{2} \)
17 \( 1 - 8.37e3T + 4.10e8T^{2} \)
19 \( 1 - 2.12e4iT - 8.93e8T^{2} \)
23 \( 1 - 1.36e3T + 3.40e9T^{2} \)
29 \( 1 + 9.65e4T + 1.72e10T^{2} \)
31 \( 1 + 1.11e5iT - 2.75e10T^{2} \)
37 \( 1 - 4.65e5iT - 9.49e10T^{2} \)
41 \( 1 + 9.74e4iT - 1.94e11T^{2} \)
43 \( 1 - 4.03e5T + 2.71e11T^{2} \)
47 \( 1 + 1.90e4iT - 5.06e11T^{2} \)
53 \( 1 - 1.14e6T + 1.17e12T^{2} \)
59 \( 1 + 2.81e6iT - 2.48e12T^{2} \)
61 \( 1 + 5.46e5T + 3.14e12T^{2} \)
67 \( 1 - 1.93e6iT - 6.06e12T^{2} \)
71 \( 1 + 1.12e6iT - 9.09e12T^{2} \)
73 \( 1 + 3.91e4iT - 1.10e13T^{2} \)
79 \( 1 + 2.19e6T + 1.92e13T^{2} \)
83 \( 1 + 9.73e6iT - 2.71e13T^{2} \)
89 \( 1 - 8.59e6iT - 4.42e13T^{2} \)
97 \( 1 + 7.31e6iT - 8.07e13T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.97558650771254353487797902676, −11.33281795419901500655195091822, −10.11573651131084291337104961009, −8.879873090343055450456757339501, −7.81055022307665857541736038132, −6.62548380647639100092500312500, −5.81028820835913716747886893152, −5.04750310732258932502984352325, −2.85810668782116824202375271308, −1.90544191033579816910801408773, 0.49292601228208192088554720595, 1.05979560112169512093227330578, 2.95611514605203479414626367568, 4.00218408797948299848612686339, 5.51908899970035118205344888297, 6.63852934129408032233767666269, 7.58434356857282800871632835310, 8.963612571143143223417522450689, 10.45794744626678238270158398336, 10.88399882792408293811121637233

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