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Properties

Label 2-819-91.4-c1-0-38
Degree $2$
Conductor $819$
Sign $0.393 + 0.919i$
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  + 2.04i·2-s − 2.19·4-s + (−0.341 + 0.197i)5-s + (−0.908 − 2.48i)7-s − 0.400i·8-s + (−0.403 − 0.699i)10-s + (−1.35 + 0.783i)11-s + (1.46 − 3.29i)13-s + (5.09 − 1.85i)14-s − 3.57·16-s − 6.82·17-s + (−6.75 − 3.89i)19-s + (0.749 − 0.432i)20-s + (−1.60 − 2.77i)22-s − 4.78·23-s + ⋯
L(s)  = 1  + 1.44i·2-s − 1.09·4-s + (−0.152 + 0.0881i)5-s + (−0.343 − 0.939i)7-s − 0.141i·8-s + (−0.127 − 0.221i)10-s + (−0.409 + 0.236i)11-s + (0.406 − 0.913i)13-s + (1.36 − 0.497i)14-s − 0.892·16-s − 1.65·17-s + (−1.54 − 0.894i)19-s + (0.167 − 0.0967i)20-s + (−0.342 − 0.592i)22-s − 0.997·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.393 + 0.919i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.393 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.393 + 0.919i$
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: \((2,\ 819,\ (\ :1/2),\ 0.393 + 0.919i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.193492 - 0.127628i\)
\(L(\frac12)\) \(\approx\) \(0.193492 - 0.127628i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.908 + 2.48i)T \)
13 \( 1 + (-1.46 + 3.29i)T \)
good2 \( 1 - 2.04iT - 2T^{2} \)
5 \( 1 + (0.341 - 0.197i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.35 - 0.783i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + 6.82T + 17T^{2} \)
19 \( 1 + (6.75 + 3.89i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 4.78T + 23T^{2} \)
29 \( 1 + (-3.94 + 6.82i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.14 - 2.97i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 1.17iT - 37T^{2} \)
41 \( 1 + (6.17 + 3.56i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.65 - 6.32i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.92 + 2.84i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.964 + 1.67i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 11.4iT - 59T^{2} \)
61 \( 1 + (-6.13 + 10.6i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.41 - 1.39i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (7.66 - 4.42i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.23 + 1.29i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.55 - 2.69i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.01iT - 83T^{2} \)
89 \( 1 + 12.4iT - 89T^{2} \)
97 \( 1 + (9.58 - 5.53i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.04732366077056070874292448671, −8.822067591488811016390882846792, −8.215354004135961129090201660018, −7.38567356440854145525391939994, −6.61395243921868668804433696269, −6.03564602411370797490826065001, −4.76057698739550058306783803099, −4.08291115372844044276729896327, −2.46064160705265002974743468976, −0.10493235305824068035283426411, 1.90530484204167521084291102449, 2.56414129744727827816330984178, 3.88792706695073178551061195773, 4.56040480192791424986806795789, 6.06504596024589131695494291438, 6.71601040036812268177882522258, 8.427323597496065233378470331445, 8.754537474436166187102352024254, 9.767595834381867501515764011353, 10.53505658987543400418865055596

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