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Properties

Label 2-819-91.24-c1-0-28
Degree $2$
Conductor $819$
Sign $0.976 - 0.215i$
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  + (1.46 − 0.393i)2-s + (0.271 − 0.156i)4-s + (3.59 + 0.962i)5-s + (2.64 + 0.145i)7-s + (−1.81 + 1.81i)8-s + 5.65·10-s + (1.41 − 1.41i)11-s + (−1.45 + 3.29i)13-s + (3.93 − 0.825i)14-s + (−2.26 + 3.92i)16-s + (−2.36 − 4.08i)17-s + (−3.15 + 3.15i)19-s + (1.12 − 0.302i)20-s + (1.51 − 2.62i)22-s + (1.80 + 1.04i)23-s + ⋯
L(s)  = 1  + (1.03 − 0.278i)2-s + (0.135 − 0.0784i)4-s + (1.60 + 0.430i)5-s + (0.998 + 0.0551i)7-s + (−0.641 + 0.641i)8-s + 1.78·10-s + (0.425 − 0.425i)11-s + (−0.404 + 0.914i)13-s + (1.05 − 0.220i)14-s + (−0.566 + 0.980i)16-s + (−0.572 − 0.991i)17-s + (−0.723 + 0.723i)19-s + (0.252 − 0.0675i)20-s + (0.323 − 0.560i)22-s + (0.377 + 0.217i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.215i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.976 - 0.215i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.976 - 0.215i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (388, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.976 - 0.215i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.27397 + 0.356842i\)
\(L(\frac12)\) \(\approx\) \(3.27397 + 0.356842i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.64 - 0.145i)T \)
13 \( 1 + (1.45 - 3.29i)T \)
good2 \( 1 + (-1.46 + 0.393i)T + (1.73 - i)T^{2} \)
5 \( 1 + (-3.59 - 0.962i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-1.41 + 1.41i)T - 11iT^{2} \)
17 \( 1 + (2.36 + 4.08i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.15 - 3.15i)T - 19iT^{2} \)
23 \( 1 + (-1.80 - 1.04i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (5.10 + 8.85i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.591 + 2.20i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-0.166 - 0.622i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-8.81 - 2.36i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-0.0966 - 0.0557i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.16 + 4.33i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-2.18 + 3.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.438 - 1.63i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + 6.75iT - 61T^{2} \)
67 \( 1 + (2.10 + 2.10i)T + 67iT^{2} \)
71 \( 1 + (14.4 - 3.87i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (10.1 - 2.72i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (0.0273 + 0.0473i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-9.05 + 9.05i)T - 83iT^{2} \)
89 \( 1 + (-6.91 + 1.85i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (3.20 + 11.9i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.37237488527467603299084844685, −9.351012059209444590151052012197, −8.865924156149675226506094621533, −7.57563551488479229815827171213, −6.35607280921954976563978117278, −5.77866215644874649455976947057, −4.86709097197202904414660824172, −4.02849619757018909878669910651, −2.55858626148677282363077984965, −1.90495108198258116399029325639, 1.43149000174447135837768099062, 2.61581045412028006712061378805, 4.18463300716435726625347923180, 4.98069360867343044179483700724, 5.60180880590819587066275856310, 6.37589617085535517954055936073, 7.37318970114621470650654271166, 8.840558229366013280843718700161, 9.145561473952556652536632398855, 10.32738254557867183401369403031

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