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Properties

Label 2-819-91.20-c1-0-42
Degree $2$
Conductor $819$
Sign $0.390 + 0.920i$
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  + (2.47 − 0.664i)2-s + (3.97 − 2.29i)4-s + (−0.281 + 0.281i)5-s + (1.14 − 2.38i)7-s + (4.70 − 4.70i)8-s + (−0.511 + 0.885i)10-s + (−0.939 − 3.50i)11-s + (−3.44 + 1.06i)13-s + (1.25 − 6.67i)14-s + (3.95 − 6.84i)16-s + (2.04 + 3.54i)17-s + (−0.777 − 0.208i)19-s + (−0.473 + 1.76i)20-s + (−4.66 − 8.07i)22-s + (4.41 + 2.54i)23-s + ⋯
L(s)  = 1  + (1.75 − 0.469i)2-s + (1.98 − 1.14i)4-s + (−0.125 + 0.125i)5-s + (0.432 − 0.901i)7-s + (1.66 − 1.66i)8-s + (−0.161 + 0.280i)10-s + (−0.283 − 1.05i)11-s + (−0.955 + 0.294i)13-s + (0.334 − 1.78i)14-s + (0.987 − 1.71i)16-s + (0.496 + 0.860i)17-s + (−0.178 − 0.0478i)19-s + (−0.105 + 0.395i)20-s + (−0.993 − 1.72i)22-s + (0.919 + 0.531i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.390 + 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.390 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.390 + 0.920i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (748, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.390 + 0.920i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.54457 - 2.34714i\)
\(L(\frac12)\) \(\approx\) \(3.54457 - 2.34714i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.14 + 2.38i)T \)
13 \( 1 + (3.44 - 1.06i)T \)
good2 \( 1 + (-2.47 + 0.664i)T + (1.73 - i)T^{2} \)
5 \( 1 + (0.281 - 0.281i)T - 5iT^{2} \)
11 \( 1 + (0.939 + 3.50i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-2.04 - 3.54i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.777 + 0.208i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-4.41 - 2.54i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.00 - 1.74i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.44 + 4.44i)T - 31iT^{2} \)
37 \( 1 + (-0.463 - 1.73i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-0.578 - 2.15i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (2.65 - 1.53i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.99 - 5.99i)T + 47iT^{2} \)
53 \( 1 + 9.09T + 53T^{2} \)
59 \( 1 + (-1.92 + 7.17i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (2.40 - 1.38i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.85 - 1.30i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-0.582 + 2.17i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-3.50 - 3.50i)T + 73iT^{2} \)
79 \( 1 + 3.61T + 79T^{2} \)
83 \( 1 + (-5.36 + 5.36i)T - 83iT^{2} \)
89 \( 1 + (11.5 - 3.09i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-7.71 - 2.06i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.59348766181486854603429931399, −9.554667102291695779542368169553, −8.072262154822094477627104201023, −7.23847205495835073490085710830, −6.31148173325545092411403976401, −5.38611428965650413371473425337, −4.59489080528082823323484938661, −3.67484069556054097760891480080, −2.85589308747747484921751531131, −1.41503783297947911936204966224, 2.25878165556941291273079364405, 2.97364647842541230251703318066, 4.46645739517176250482336022238, 4.94516606661474861801907830324, 5.69769095573297534527957657174, 6.80774466329675424637326472000, 7.46551810892365025568222868279, 8.379712916044770968942064737728, 9.598105016313919610304628462019, 10.65045239564158409201865192077

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