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Properties

Label 2-13e2-13.12-c7-0-36
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 21.7i·2-s − 67.8·3-s − 346.·4-s − 157. i·5-s + 1.47e3i·6-s + 842. i·7-s + 4.77e3i·8-s + 2.41e3·9-s − 3.43e3·10-s − 1.99e3i·11-s + 2.35e4·12-s + 1.83e4·14-s + 1.07e4i·15-s + 5.96e4·16-s − 9.62e3·17-s − 5.26e4i·18-s + ⋯
L(s)  = 1  − 1.92i·2-s − 1.45·3-s − 2.71·4-s − 0.564i·5-s + 2.79i·6-s + 0.927i·7-s + 3.29i·8-s + 1.10·9-s − 1.08·10-s − 0.451i·11-s + 3.93·12-s + 1.78·14-s + 0.818i·15-s + 3.63·16-s − 0.474·17-s − 2.12i·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

\(L(4)\) \(\approx\) \(0.5063389739\)
\(L(\frac12)\) \(\approx\) \(0.5063389739\)
\(L(\frac{9}{2})\) 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 \( 1 + 21.7iT - 128T^{2} \)
3 \( 1 + 67.8T + 2.18e3T^{2} \)
5 \( 1 + 157. iT - 7.81e4T^{2} \)
7 \( 1 - 842. iT - 8.23e5T^{2} \)
11 \( 1 + 1.99e3iT - 1.94e7T^{2} \)
17 \( 1 + 9.62e3T + 4.10e8T^{2} \)
19 \( 1 - 8.43e3iT - 8.93e8T^{2} \)
23 \( 1 + 3.41e4T + 3.40e9T^{2} \)
29 \( 1 + 9.67e3T + 1.72e10T^{2} \)
31 \( 1 + 2.05e5iT - 2.75e10T^{2} \)
37 \( 1 - 4.32e5iT - 9.49e10T^{2} \)
41 \( 1 + 1.13e5iT - 1.94e11T^{2} \)
43 \( 1 - 4.00e5T + 2.71e11T^{2} \)
47 \( 1 - 1.32e6iT - 5.06e11T^{2} \)
53 \( 1 - 2.98e5T + 1.17e12T^{2} \)
59 \( 1 + 1.63e6iT - 2.48e12T^{2} \)
61 \( 1 - 1.66e6T + 3.14e12T^{2} \)
67 \( 1 + 1.42e6iT - 6.06e12T^{2} \)
71 \( 1 - 1.03e6iT - 9.09e12T^{2} \)
73 \( 1 - 4.43e6iT - 1.10e13T^{2} \)
79 \( 1 + 7.40e6T + 1.92e13T^{2} \)
83 \( 1 + 7.53e5iT - 2.71e13T^{2} \)
89 \( 1 + 2.49e6iT - 4.42e13T^{2} \)
97 \( 1 + 6.38e6iT - 8.07e13T^{2} \)
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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

−11.22486633144736477578055809827, −10.23966474893366742941612405084, −9.262448285446477422502427477381, −8.321474831841921171922904510742, −6.01981061881754087743380353000, −5.15953638643277826779806251783, −4.21512762021340405202261159426, −2.63889412385859513178685277633, −1.30073364634274132939900258404, −0.28921127799695434504499636790, 0.71934860822066789842772638971, 4.01903669750111846183870833477, 4.96894516058478244759720637116, 5.92506626883978693998823645193, 6.91807077785162149522162078214, 7.25411319092295426565125530199, 8.723560789737594595956306584822, 10.06943350108820541086616860566, 10.82013289829489040070686441068, 12.26592280105850207415038410538

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