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Properties

Label 2-269-269.268-c3-0-17
Degree $2$
Conductor $269$
Sign $0.171 + 0.985i$
Analytic cond. $15.8715$
Root an. cond. $3.98390$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.45i·2-s + 0.747i·3-s − 11.8·4-s − 16.4·5-s + 3.33·6-s + 20.0i·7-s + 17.2i·8-s + 26.4·9-s + 73.3i·10-s + 19.7·11-s − 8.87i·12-s + 53.8·13-s + 89.3·14-s − 12.2i·15-s − 17.9·16-s + 79.4i·17-s + ⋯
L(s)  = 1  − 1.57i·2-s + 0.143i·3-s − 1.48·4-s − 1.47·5-s + 0.226·6-s + 1.08i·7-s + 0.764i·8-s + 0.979·9-s + 2.31i·10-s + 0.542·11-s − 0.213i·12-s + 1.14·13-s + 1.70·14-s − 0.211i·15-s − 0.280·16-s + 1.13i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 269 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.171 + 0.985i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 269 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.171 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(269\)
Sign: $0.171 + 0.985i$
Analytic conductor: \(15.8715\)
Root analytic conductor: \(3.98390\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{269} (268, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 269,\ (\ :3/2),\ 0.171 + 0.985i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.384159244\)
\(L(\frac12)\) \(\approx\) \(1.384159244\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad269 \( 1 + (-756. - 4.34e3i)T \)
good2 \( 1 + 4.45iT - 8T^{2} \)
3 \( 1 - 0.747iT - 27T^{2} \)
5 \( 1 + 16.4T + 125T^{2} \)
7 \( 1 - 20.0iT - 343T^{2} \)
11 \( 1 - 19.7T + 1.33e3T^{2} \)
13 \( 1 - 53.8T + 2.19e3T^{2} \)
17 \( 1 - 79.4iT - 4.91e3T^{2} \)
19 \( 1 + 108. iT - 6.85e3T^{2} \)
23 \( 1 + 140.T + 1.21e4T^{2} \)
29 \( 1 - 32.4iT - 2.43e4T^{2} \)
31 \( 1 + 174. iT - 2.97e4T^{2} \)
37 \( 1 - 391.T + 5.06e4T^{2} \)
41 \( 1 - 234.T + 6.89e4T^{2} \)
43 \( 1 - 458.T + 7.95e4T^{2} \)
47 \( 1 - 522.T + 1.03e5T^{2} \)
53 \( 1 + 235.T + 1.48e5T^{2} \)
59 \( 1 - 714. iT - 2.05e5T^{2} \)
61 \( 1 - 541.T + 2.26e5T^{2} \)
67 \( 1 + 410.T + 3.00e5T^{2} \)
71 \( 1 + 1.04e3iT - 3.57e5T^{2} \)
73 \( 1 + 284.T + 3.89e5T^{2} \)
79 \( 1 - 1.06e3T + 4.93e5T^{2} \)
83 \( 1 - 1.20e3iT - 5.71e5T^{2} \)
89 \( 1 + 884.T + 7.04e5T^{2} \)
97 \( 1 + 738.T + 9.12e5T^{2} \)
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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

−11.27096932194834042724351002532, −10.75401751335313487066188723175, −9.455577284756767751587007811507, −8.743136498046100062050215225159, −7.64802353016456202006360539472, −6.13038642920382694452861830122, −4.21328985194815750309766540367, −3.94783182728861186260305335491, −2.45520390050506819949706652213, −0.934905016698655151829667968438, 0.815002802326285905624204778907, 3.95489491357263077066616055393, 4.28845076358571142929856166172, 5.98083465341125727419863235077, 7.02415973482067021653285139634, 7.62496230235739027922646498284, 8.228753672623031620972726573139, 9.496433040642095908452546161202, 10.76345689096477283820652840229, 11.75541247443568772419755746077

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