Properties

Label 2-819-91.54-c1-0-12
Degree $2$
Conductor $819$
Sign $-0.422 - 0.906i$
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  + (−1.55 + 1.55i)2-s − 2.84i·4-s + (−3.12 + 0.837i)5-s + (1.93 + 1.79i)7-s + (1.31 + 1.31i)8-s + (3.56 − 6.16i)10-s + (3.89 − 1.04i)11-s + (−2.61 − 2.47i)13-s + (−5.81 + 0.218i)14-s + 1.59·16-s + 7.61·17-s + (−0.714 + 2.66i)19-s + (2.38 + 8.89i)20-s + (−4.43 + 7.68i)22-s − 4.49i·23-s + ⋯
L(s)  = 1  + (−1.10 + 1.10i)2-s − 1.42i·4-s + (−1.39 + 0.374i)5-s + (0.733 + 0.680i)7-s + (0.465 + 0.465i)8-s + (1.12 − 1.95i)10-s + (1.17 − 0.314i)11-s + (−0.726 − 0.687i)13-s + (−1.55 + 0.0584i)14-s + 0.398·16-s + 1.84·17-s + (−0.163 + 0.611i)19-s + (0.532 + 1.98i)20-s + (−0.946 + 1.63i)22-s − 0.937i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.422 - 0.906i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.422 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.422 - 0.906i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.422 - 0.906i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.377722 + 0.592941i\)
\(L(\frac12)\) \(\approx\) \(0.377722 + 0.592941i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.93 - 1.79i)T \)
13 \( 1 + (2.61 + 2.47i)T \)
good2 \( 1 + (1.55 - 1.55i)T - 2iT^{2} \)
5 \( 1 + (3.12 - 0.837i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-3.89 + 1.04i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 - 7.61T + 17T^{2} \)
19 \( 1 + (0.714 - 2.66i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + 4.49iT - 23T^{2} \)
29 \( 1 + (1.49 + 2.58i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.691 + 2.58i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-1.20 - 1.20i)T + 37iT^{2} \)
41 \( 1 + (2.95 - 11.0i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-6.30 - 3.64i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.666 - 2.48i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-3.91 - 6.77i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.78 + 5.78i)T - 59iT^{2} \)
61 \( 1 + (7.97 - 4.60i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.02 - 7.54i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-0.721 - 2.69i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-1.69 - 0.453i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-4.31 + 7.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.63 + 1.63i)T + 83iT^{2} \)
89 \( 1 + (12.7 - 12.7i)T - 89iT^{2} \)
97 \( 1 + (-12.7 + 3.41i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.23117388639004419083330308842, −9.453296503267881356118857798935, −8.448345184948759819066107457267, −7.911989267702365619574086779923, −7.49009527016840227125920430234, −6.36627780713291296666406982574, −5.56330354432004196918732218260, −4.26290351854680622052659704977, −3.07637802556255871501504509908, −1.00116868907132078807385865796, 0.71209858331307465581361165292, 1.77348136823683599491461230384, 3.46531838575248628090559825900, 4.07676241734879963367318092996, 5.24055673998124691629686047541, 7.17166515731072582223114259726, 7.54523833465925184586050145991, 8.467642980372915377588010615560, 9.175189271566126838394893051854, 9.981751179667570725233837298805

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