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Properties

Label 2-819-117.20-c1-0-65
Degree $2$
Conductor $819$
Sign $0.114 + 0.993i$
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  + (1.17 + 0.315i)2-s + (−0.0778 + 1.73i)3-s + (−0.445 − 0.257i)4-s + (−1.47 − 0.395i)5-s + (−0.637 + 2.01i)6-s + (0.707 − 0.707i)7-s + (−2.16 − 2.16i)8-s + (−2.98 − 0.269i)9-s + (−1.61 − 0.931i)10-s + (0.102 − 0.383i)11-s + (0.479 − 0.750i)12-s + (−0.330 − 3.59i)13-s + (1.05 − 0.609i)14-s + (0.799 − 2.52i)15-s + (−1.35 − 2.34i)16-s + (−1.67 − 2.90i)17-s + ⋯
L(s)  = 1  + (0.832 + 0.223i)2-s + (−0.0449 + 0.998i)3-s + (−0.222 − 0.128i)4-s + (−0.659 − 0.176i)5-s + (−0.260 + 0.821i)6-s + (0.267 − 0.267i)7-s + (−0.766 − 0.766i)8-s + (−0.995 − 0.0897i)9-s + (−0.510 − 0.294i)10-s + (0.0309 − 0.115i)11-s + (0.138 − 0.216i)12-s + (−0.0916 − 0.995i)13-s + (0.282 − 0.162i)14-s + (0.206 − 0.651i)15-s + (−0.338 − 0.586i)16-s + (−0.406 − 0.703i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.114 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.114 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.114 + 0.993i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (722, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.114 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.702585 - 0.626149i\)
\(L(\frac12)\) \(\approx\) \(0.702585 - 0.626149i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.0778 - 1.73i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
13 \( 1 + (0.330 + 3.59i)T \)
good2 \( 1 + (-1.17 - 0.315i)T + (1.73 + i)T^{2} \)
5 \( 1 + (1.47 + 0.395i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.102 + 0.383i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (1.67 + 2.90i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.227 - 0.847i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 - 2.89T + 23T^{2} \)
29 \( 1 + (5.53 - 3.19i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.69 + 10.0i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-1.16 - 4.33i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-4.21 + 4.21i)T - 41iT^{2} \)
43 \( 1 + 1.67iT - 43T^{2} \)
47 \( 1 + (8.80 - 2.35i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 - 1.65iT - 53T^{2} \)
59 \( 1 + (-2.04 + 0.549i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + 8.25T + 61T^{2} \)
67 \( 1 + (0.544 + 0.544i)T + 67iT^{2} \)
71 \( 1 + (4.37 + 1.17i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-3.05 + 3.05i)T - 73iT^{2} \)
79 \( 1 + (1.00 - 1.73i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.617 - 2.30i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + (11.1 - 2.97i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-1.45 - 1.45i)T + 97iT^{2} \)
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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

−9.969824440337027564466716945879, −9.325925169050362500990821799724, −8.384226519525293124704824349131, −7.49586029850573170333170448878, −6.16906001469254173315438692555, −5.35809438769237709867111884828, −4.58187151824547003742274078771, −3.88804500653585667394659820157, −2.94887776430903648590065574607, −0.35097269485878259693380578169, 1.82619602814467202263022230880, 3.00635277746602206328782166452, 4.05293448708019747806964220551, 5.00009842486541935842621142672, 6.02878055889007269047510651244, 6.91623505737968664489802952906, 7.84660761591087272147095707565, 8.602313141101911793775713334092, 9.333174368963460174663536070049, 10.93332058317104408898643942561

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