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Properties

Label 2-819-117.50-c1-0-17
Degree $2$
Conductor $819$
Sign $0.462 + 0.886i$
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  + (−0.727 − 2.71i)2-s + (1.68 + 0.410i)3-s + (−5.10 + 2.94i)4-s + (−0.0876 − 0.327i)5-s + (−0.110 − 4.86i)6-s + (−0.707 + 0.707i)7-s + (7.73 + 7.73i)8-s + (2.66 + 1.38i)9-s + (−0.824 + 0.475i)10-s + (−0.128 + 0.0343i)11-s + (−9.80 + 2.86i)12-s + (−3.19 + 1.67i)13-s + (2.43 + 1.40i)14-s + (−0.0132 − 0.586i)15-s + (9.48 − 16.4i)16-s + (−2.30 + 3.98i)17-s + ⋯
L(s)  = 1  + (−0.514 − 1.91i)2-s + (0.971 + 0.236i)3-s + (−2.55 + 1.47i)4-s + (−0.0392 − 0.146i)5-s + (−0.0449 − 1.98i)6-s + (−0.267 + 0.267i)7-s + (2.73 + 2.73i)8-s + (0.887 + 0.460i)9-s + (−0.260 + 0.150i)10-s + (−0.0386 + 0.0103i)11-s + (−2.82 + 0.827i)12-s + (−0.884 + 0.465i)13-s + (0.650 + 0.375i)14-s + (−0.00342 − 0.151i)15-s + (2.37 − 4.10i)16-s + (−0.558 + 0.967i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.462 + 0.886i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.462 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.462 + 0.886i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (50, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.462 + 0.886i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12490 - 0.681759i\)
\(L(\frac12)\) \(\approx\) \(1.12490 - 0.681759i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.68 - 0.410i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
13 \( 1 + (3.19 - 1.67i)T \)
good2 \( 1 + (0.727 + 2.71i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (0.0876 + 0.327i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.128 - 0.0343i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (2.30 - 3.98i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.40 + 0.375i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 - 6.68T + 23T^{2} \)
29 \( 1 + (-8.80 - 5.08i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.93 + 1.05i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-1.96 - 0.525i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (3.40 - 3.40i)T - 41iT^{2} \)
43 \( 1 - 0.860iT - 43T^{2} \)
47 \( 1 + (-2.32 + 8.69i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 - 9.59iT - 53T^{2} \)
59 \( 1 + (1.80 - 6.73i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + 10.5T + 61T^{2} \)
67 \( 1 + (-3.63 - 3.63i)T + 67iT^{2} \)
71 \( 1 + (-0.110 - 0.411i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-1.88 + 1.88i)T - 73iT^{2} \)
79 \( 1 + (1.25 + 2.16i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.16 + 1.38i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + (-2.28 + 8.52i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (9.52 + 9.52i)T + 97iT^{2} \)
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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

−10.24788082647736266989927716238, −9.254934396202649399296399632962, −8.824953782857731302064980176079, −8.134335783981456689133510796061, −7.01392497322836983742624475911, −4.89186311376335954050516148486, −4.34733540298012252129146972637, −3.11041822350661265498055353236, −2.56016208601177302703514788378, −1.30817998086726329883730877608, 0.828544386755051177415372443936, 2.99127103270181037920506863269, 4.46272509409500273853946468958, 5.14333531582579976219714409398, 6.55851614834668734201935563525, 6.99157201841927133451948671613, 7.75114076623243910162039698515, 8.449007499968724221217685567354, 9.324240359592047051538160102165, 9.769154597760176028316232716330

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