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Properties

Label 2-819-91.59-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: $\chi_{819} (514, \cdot )$
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

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

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