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Properties

Label 2-819-91.81-c1-0-34
Degree $2$
Conductor $819$
Sign $-0.245 + 0.969i$
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  + (0.929 − 1.60i)2-s + (−0.726 − 1.25i)4-s + (−0.0986 − 0.170i)5-s + (1.58 − 2.11i)7-s + 1.01·8-s − 0.366·10-s + 4.18·11-s + (−2.72 + 2.36i)13-s + (−1.92 − 4.52i)14-s + (2.39 − 4.15i)16-s + (0.420 + 0.728i)17-s + 1.35·19-s + (−0.143 + 0.248i)20-s + (3.88 − 6.73i)22-s + (−2.05 + 3.56i)23-s + ⋯
L(s)  = 1  + (0.656 − 1.13i)2-s + (−0.363 − 0.629i)4-s + (−0.0441 − 0.0764i)5-s + (0.600 − 0.799i)7-s + 0.359·8-s − 0.115·10-s + 1.26·11-s + (−0.755 + 0.655i)13-s + (−0.515 − 1.20i)14-s + (0.599 − 1.03i)16-s + (0.102 + 0.176i)17-s + 0.310·19-s + (−0.0320 + 0.0555i)20-s + (0.828 − 1.43i)22-s + (−0.429 + 0.743i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.245 + 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.245 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.245 + 0.969i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.245 + 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55376 - 1.99533i\)
\(L(\frac12)\) \(\approx\) \(1.55376 - 1.99533i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.58 + 2.11i)T \)
13 \( 1 + (2.72 - 2.36i)T \)
good2 \( 1 + (-0.929 + 1.60i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (0.0986 + 0.170i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 - 4.18T + 11T^{2} \)
17 \( 1 + (-0.420 - 0.728i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 1.35T + 19T^{2} \)
23 \( 1 + (2.05 - 3.56i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.11 + 7.13i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.640 + 1.10i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.52 - 2.63i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.69 - 4.67i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.66 - 4.61i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.83 + 10.1i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.32 + 4.02i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.02 - 5.24i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 11.3T + 61T^{2} \)
67 \( 1 - 13.3T + 67T^{2} \)
71 \( 1 + (2.98 - 5.17i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.94 - 3.36i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.36 - 9.29i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.07T + 83T^{2} \)
89 \( 1 + (5.99 - 10.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (9.73 - 16.8i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.03527554390881209628375505254, −9.608513280933738473693054827554, −8.282382092885181906353011813198, −7.41248533529890451509701460910, −6.50069402136638557573487940910, −5.10500839243738070052083832854, −4.24800414310575748154810176266, −3.67703022100588445014870901682, −2.24742214560090678923307671049, −1.20939368572512834651453559466, 1.68518776984085736790988006046, 3.28605299131730205016299455967, 4.53091582492916252969303853551, 5.27790845361970450419751487407, 6.02940862186533919466313394620, 6.99061725325893658493230969445, 7.64065579794201520993842704509, 8.634215168271506778359448313607, 9.363274421082688712104446843009, 10.54775858019867675296852509230

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