Properties

Label 2-819-117.2-c1-0-3
Degree $2$
Conductor $819$
Sign $0.448 - 0.893i$
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  + (−0.653 − 0.653i)2-s + (0.319 − 1.70i)3-s − 1.14i·4-s + (−0.561 + 2.09i)5-s + (−1.32 + 0.903i)6-s + (−0.258 + 0.965i)7-s + (−2.05 + 2.05i)8-s + (−2.79 − 1.08i)9-s + (1.73 − 1.00i)10-s + (1.77 − 1.77i)11-s + (−1.94 − 0.366i)12-s + (−2.26 + 2.80i)13-s + (0.800 − 0.462i)14-s + (3.38 + 1.62i)15-s + 0.397·16-s + (−2.30 + 3.99i)17-s + ⋯
L(s)  = 1  + (−0.462 − 0.462i)2-s + (0.184 − 0.982i)3-s − 0.572i·4-s + (−0.251 + 0.937i)5-s + (−0.539 + 0.368i)6-s + (−0.0978 + 0.365i)7-s + (−0.726 + 0.726i)8-s + (−0.931 − 0.362i)9-s + (0.549 − 0.317i)10-s + (0.533 − 0.533i)11-s + (−0.562 − 0.105i)12-s + (−0.628 + 0.778i)13-s + (0.213 − 0.123i)14-s + (0.875 + 0.419i)15-s + 0.0993·16-s + (−0.559 + 0.968i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.448 - 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.448 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.448 - 0.893i$
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: \((2,\ 819,\ (\ :1/2),\ 0.448 - 0.893i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.343158 + 0.211722i\)
\(L(\frac12)\) \(\approx\) \(0.343158 + 0.211722i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.319 + 1.70i)T \)
7 \( 1 + (0.258 - 0.965i)T \)
13 \( 1 + (2.26 - 2.80i)T \)
good2 \( 1 + (0.653 + 0.653i)T + 2iT^{2} \)
5 \( 1 + (0.561 - 2.09i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-1.77 + 1.77i)T - 11iT^{2} \)
17 \( 1 + (2.30 - 3.99i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.13 + 4.24i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (3.48 - 6.04i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.42iT - 29T^{2} \)
31 \( 1 + (5.78 + 1.55i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (1.75 - 6.55i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-1.68 + 0.451i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-2.83 + 1.63i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.16 + 8.08i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 - 13.1iT - 53T^{2} \)
59 \( 1 + (-3.51 + 3.51i)T - 59iT^{2} \)
61 \( 1 + (3.52 + 6.10i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.67 + 13.7i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-8.76 + 2.34i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-3.96 - 3.96i)T + 73iT^{2} \)
79 \( 1 + (-0.130 + 0.226i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.05 - 1.35i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-10.6 - 2.86i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (9.48 + 2.54i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.58105578405996745116909181371, −9.323226562024894737178624014747, −8.936807375707923182307099372631, −7.85149309275767836740786734691, −6.79553971631482168064141177161, −6.34735624784518672410810741952, −5.28755461314240673859359016912, −3.59322118515027974690065704267, −2.49115672702509321796806600205, −1.59786956678343213180207754098, 0.21816158657485037216948519306, 2.60075523476151090198725718738, 3.92972670521719083197528681432, 4.46205836311566299317872835933, 5.58117718529731207913771824484, 6.80342601328872160498967216688, 7.81402866534571309997976225452, 8.403123441745797425925840065840, 9.222586980257280387568760921092, 9.775020394000390277562773220837

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