Properties

Label 2-819-91.6-c1-0-19
Degree $2$
Conductor $819$
Sign $0.747 - 0.664i$
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  + (0.189 + 0.707i)2-s + (1.26 − 0.731i)4-s + (1.23 + 1.23i)5-s + (−2.64 − 0.0736i)7-s + (1.79 + 1.79i)8-s + (−0.639 + 1.10i)10-s + (3.18 − 0.854i)11-s + (2.53 − 2.56i)13-s + (−0.449 − 1.88i)14-s + (0.532 − 0.923i)16-s + (0.433 + 0.751i)17-s + (−1.01 + 3.80i)19-s + (2.46 + 0.660i)20-s + (1.21 + 2.09i)22-s + (3.77 + 2.17i)23-s + ⋯
L(s)  = 1  + (0.134 + 0.500i)2-s + (0.633 − 0.365i)4-s + (0.551 + 0.551i)5-s + (−0.999 − 0.0278i)7-s + (0.634 + 0.634i)8-s + (−0.202 + 0.350i)10-s + (0.961 − 0.257i)11-s + (0.701 − 0.712i)13-s + (−0.120 − 0.504i)14-s + (0.133 − 0.230i)16-s + (0.105 + 0.182i)17-s + (−0.233 + 0.872i)19-s + (0.551 + 0.147i)20-s + (0.257 + 0.446i)22-s + (0.786 + 0.454i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.747 - 0.664i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.747 - 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.747 - 0.664i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (370, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.747 - 0.664i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.04061 + 0.775629i\)
\(L(\frac12)\) \(\approx\) \(2.04061 + 0.775629i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.64 + 0.0736i)T \)
13 \( 1 + (-2.53 + 2.56i)T \)
good2 \( 1 + (-0.189 - 0.707i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (-1.23 - 1.23i)T + 5iT^{2} \)
11 \( 1 + (-3.18 + 0.854i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-0.433 - 0.751i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.01 - 3.80i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-3.77 - 2.17i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.65 - 4.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.220 - 0.220i)T + 31iT^{2} \)
37 \( 1 + (3.57 - 0.957i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (1.90 - 0.509i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-9.99 + 5.76i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.68 - 3.68i)T - 47iT^{2} \)
53 \( 1 - 3.55T + 53T^{2} \)
59 \( 1 + (8.89 + 2.38i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-4.78 + 2.76i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.01 - 3.80i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (11.9 + 3.20i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-5.55 + 5.55i)T - 73iT^{2} \)
79 \( 1 + 15.7T + 79T^{2} \)
83 \( 1 + (-3.80 - 3.80i)T + 83iT^{2} \)
89 \( 1 + (-3.26 - 12.1i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-0.756 + 2.82i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.45656688066698514864535064350, −9.569245381691309197863422731603, −8.642164055012475790366421582211, −7.49292494572539482714820473910, −6.64883329128780646093936167059, −6.12160386116996528522317659652, −5.43579225115932041696372565080, −3.79683548869927700230317356616, −2.84806199754717262964852097656, −1.43659642317059652817097428548, 1.27093107659136422888329259850, 2.48062933035890020130444410197, 3.59299851738028973187056303908, 4.48317452551266482678957031308, 5.91678658316419967017086984926, 6.65824326608576861763332189055, 7.33482806978217797279344870496, 8.798230746171104248726947822637, 9.280900798919088724694529917265, 10.13145839108666175532728815955

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