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Properties

Label 2-819-117.2-c1-0-43
Degree $2$
Conductor $819$
Sign $-0.798 + 0.601i$
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  + (−0.780 − 0.780i)2-s + (−1.73 + 0.0760i)3-s − 0.780i·4-s + (−0.243 + 0.908i)5-s + (1.41 + 1.29i)6-s + (0.258 − 0.965i)7-s + (−2.17 + 2.17i)8-s + (2.98 − 0.263i)9-s + (0.899 − 0.519i)10-s + (−1.16 + 1.16i)11-s + (0.0593 + 1.35i)12-s + (3.60 + 0.0979i)13-s + (−0.956 + 0.552i)14-s + (0.352 − 1.59i)15-s + 1.82·16-s + (1.13 − 1.96i)17-s + ⋯
L(s)  = 1  + (−0.552 − 0.552i)2-s + (−0.999 + 0.0439i)3-s − 0.390i·4-s + (−0.108 + 0.406i)5-s + (0.575 + 0.527i)6-s + (0.0978 − 0.365i)7-s + (−0.767 + 0.767i)8-s + (0.996 − 0.0877i)9-s + (0.284 − 0.164i)10-s + (−0.352 + 0.352i)11-s + (0.0171 + 0.390i)12-s + (0.999 + 0.0271i)13-s + (−0.255 + 0.147i)14-s + (0.0909 − 0.410i)15-s + 0.457·16-s + (0.275 − 0.477i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.798 + 0.601i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.798 + 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.798 + 0.601i$
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: \((2,\ 819,\ (\ :1/2),\ -0.798 + 0.601i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.177931 - 0.531775i\)
\(L(\frac12)\) \(\approx\) \(0.177931 - 0.531775i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.73 - 0.0760i)T \)
7 \( 1 + (-0.258 + 0.965i)T \)
13 \( 1 + (-3.60 - 0.0979i)T \)
good2 \( 1 + (0.780 + 0.780i)T + 2iT^{2} \)
5 \( 1 + (0.243 - 0.908i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (1.16 - 1.16i)T - 11iT^{2} \)
17 \( 1 + (-1.13 + 1.96i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.20 + 4.49i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-1.30 + 2.26i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.75iT - 29T^{2} \)
31 \( 1 + (0.0945 + 0.0253i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-2.83 + 10.5i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (7.06 - 1.89i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (9.75 - 5.63i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.95 + 11.0i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 - 1.62iT - 53T^{2} \)
59 \( 1 + (-1.63 + 1.63i)T - 59iT^{2} \)
61 \( 1 + (1.83 + 3.17i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.74 + 13.9i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-10.8 + 2.89i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (10.5 + 10.5i)T + 73iT^{2} \)
79 \( 1 + (3.33 - 5.77i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (14.1 - 3.78i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-0.492 - 0.131i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-1.40 - 0.377i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.11501420223068890968666778689, −9.341827257635286233138639542229, −8.372738019941184135626603826225, −7.11342872724938940075274016518, −6.48805254456546066420120718630, −5.41975129187110751496759506213, −4.65313532094327804872872069177, −3.21005883018530438242758287798, −1.72881559458207863163701855964, −0.42995770601102699671713204385, 1.28527254468771297070916853274, 3.28400257819638051645369085178, 4.37714922951231267409735508587, 5.57407885701194549339260581487, 6.24511864933582691569124389349, 7.08996483708386553617002102765, 8.271714371178162498307244148577, 8.470558728407856191992169816127, 9.747809299770222168524657417729, 10.45425959811236562909852951048

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