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Properties

Label 2-819-117.2-c1-0-36
Degree $2$
Conductor $819$
Sign $0.00499 + 0.999i$
Analytic cond. $6.53974$
Root an. cond. $2.55729$
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  + (−1.87 − 1.87i)2-s + (0.832 + 1.51i)3-s + 5.01i·4-s + (0.0815 − 0.304i)5-s + (1.28 − 4.40i)6-s + (0.258 − 0.965i)7-s + (5.64 − 5.64i)8-s + (−1.61 + 2.52i)9-s + (−0.723 + 0.417i)10-s + (3.03 − 3.03i)11-s + (−7.61 + 4.17i)12-s + (−3.60 − 0.0453i)13-s + (−2.29 + 1.32i)14-s + (0.530 − 0.129i)15-s − 11.1·16-s + (3.01 − 5.21i)17-s + ⋯
L(s)  = 1  + (−1.32 − 1.32i)2-s + (0.480 + 0.876i)3-s + 2.50i·4-s + (0.0364 − 0.136i)5-s + (0.524 − 1.79i)6-s + (0.0978 − 0.365i)7-s + (1.99 − 1.99i)8-s + (−0.537 + 0.843i)9-s + (−0.228 + 0.132i)10-s + (0.915 − 0.915i)11-s + (−2.19 + 1.20i)12-s + (−0.999 − 0.0125i)13-s + (−0.613 + 0.353i)14-s + (0.136 − 0.0334i)15-s − 2.78·16-s + (0.730 − 1.26i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00499 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00499 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.00499 + 0.999i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (470, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.00499 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.604073 - 0.601065i\)
\(L(\frac12)\) \(\approx\) \(0.604073 - 0.601065i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.832 - 1.51i)T \)
7 \( 1 + (-0.258 + 0.965i)T \)
13 \( 1 + (3.60 + 0.0453i)T \)
good2 \( 1 + (1.87 + 1.87i)T + 2iT^{2} \)
5 \( 1 + (-0.0815 + 0.304i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-3.03 + 3.03i)T - 11iT^{2} \)
17 \( 1 + (-3.01 + 5.21i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.15 + 4.30i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (2.47 - 4.28i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 0.320iT - 29T^{2} \)
31 \( 1 + (2.33 + 0.625i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (0.308 - 1.15i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-11.3 + 3.03i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-10.1 + 5.87i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.788 + 2.94i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + 8.87iT - 53T^{2} \)
59 \( 1 + (-2.76 + 2.76i)T - 59iT^{2} \)
61 \( 1 + (0.793 + 1.37i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.74 - 6.52i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-13.8 + 3.70i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (6.13 + 6.13i)T + 73iT^{2} \)
79 \( 1 + (2.26 - 3.91i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.28 - 1.41i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-0.565 - 0.151i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-8.76 - 2.34i)T + (84.0 + 48.5i)T^{2} \)
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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

−9.858889609467304825414591461133, −9.295953485403522488812205760849, −8.864179866175709985331774604408, −7.81549979913953798292558018594, −7.13591883214916032048191505690, −5.25989280165960810777154322450, −4.07279585436216681732823313766, −3.24365715478974111298291382513, −2.32398188055680812000145369044, −0.71101303829855819222153357403, 1.23659851279985913599642470683, 2.31494615950939850739791903196, 4.33745113251919102031510539204, 5.85290601398034452850310464158, 6.32235796551771148800125759585, 7.26032961009030269588844059314, 7.82655158568742437646400715953, 8.554553416431958917783387036372, 9.351678702785853970921439574726, 9.972331164089626462056185054882

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