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Properties

Label 2-819-91.4-c1-0-13
Degree $2$
Conductor $819$
Sign $0.617 - 0.786i$
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  + 2·4-s + (−2 + 1.73i)7-s + (−1 + 3.46i)13-s + 4·16-s + (7.5 + 4.33i)19-s + (−2.5 + 4.33i)25-s + (−4 + 3.46i)28-s + (7.5 + 4.33i)31-s − 5.19i·37-s + (−4 − 6.92i)43-s + (1.00 − 6.92i)49-s + (−2 + 6.92i)52-s + (−0.5 + 0.866i)61-s + 8·64-s + (−10.5 + 6.06i)67-s + ⋯
L(s)  = 1  + 4-s + (−0.755 + 0.654i)7-s + (−0.277 + 0.960i)13-s + 16-s + (1.72 + 0.993i)19-s + (−0.5 + 0.866i)25-s + (−0.755 + 0.654i)28-s + (1.34 + 0.777i)31-s − 0.854i·37-s + (−0.609 − 1.05i)43-s + (0.142 − 0.989i)49-s + (−0.277 + 0.960i)52-s + (−0.0640 + 0.110i)61-s + 64-s + (−1.28 + 0.740i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.617 - 0.786i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.617 - 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.617 - 0.786i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (550, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.617 - 0.786i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.60651 + 0.781725i\)
\(L(\frac12)\) \(\approx\) \(1.60651 + 0.781725i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (2 - 1.73i)T \)
13 \( 1 + (1 - 3.46i)T \)
good2 \( 1 - 2T^{2} \)
5 \( 1 + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (5.5 - 9.52i)T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + (-7.5 - 4.33i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-7.5 - 4.33i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 5.19iT - 37T^{2} \)
41 \( 1 + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4 + 6.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (10.5 - 6.06i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-12 - 6.92i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.5 + 11.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + (4.5 - 2.59i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.21542044498469552153961220043, −9.661705528373869945269933876820, −8.723162376681562215409651022055, −7.59583823397693846133706873422, −6.92177530320279608055596495996, −6.03362407839614074902443083005, −5.27324062478636023442147678404, −3.69957392534034410954236123761, −2.80543961373722632397572010385, −1.61389568997971843237903016251, 0.917135504782075917951374909555, 2.66819301919653587642606797314, 3.34539734953464998389171891832, 4.76036566512188688452287553418, 5.91367053766579945903153450923, 6.67140302187334630284381890088, 7.50071049439439988675137716864, 8.142200196690630770983997233034, 9.635023765519565968415240702370, 10.03090772372242502431819257984

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