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Properties

Label 2-819-117.2-c1-0-46
Degree $2$
Conductor $819$
Sign $0.828 - 0.560i$
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  + (1.60 + 1.60i)2-s + (−1.15 − 1.29i)3-s + 3.12i·4-s + (0.0631 − 0.235i)5-s + (0.220 − 3.91i)6-s + (0.258 − 0.965i)7-s + (−1.80 + 1.80i)8-s + (−0.337 + 2.98i)9-s + (0.478 − 0.276i)10-s + (4.20 − 4.20i)11-s + (4.03 − 3.60i)12-s + (0.439 + 3.57i)13-s + (1.96 − 1.13i)14-s + (−0.377 + 0.190i)15-s + 0.478·16-s + (−0.508 + 0.880i)17-s + ⋯
L(s)  = 1  + (1.13 + 1.13i)2-s + (−0.666 − 0.745i)3-s + 1.56i·4-s + (0.0282 − 0.105i)5-s + (0.0902 − 1.59i)6-s + (0.0978 − 0.365i)7-s + (−0.637 + 0.637i)8-s + (−0.112 + 0.993i)9-s + (0.151 − 0.0873i)10-s + (1.26 − 1.26i)11-s + (1.16 − 1.04i)12-s + (0.121 + 0.992i)13-s + (0.524 − 0.302i)14-s + (−0.0974 + 0.0491i)15-s + 0.119·16-s + (−0.123 + 0.213i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.828 - 0.560i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.828 - 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.828 - 0.560i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (470, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.828 - 0.560i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.38247 + 0.729788i\)
\(L(\frac12)\) \(\approx\) \(2.38247 + 0.729788i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.15 + 1.29i)T \)
7 \( 1 + (-0.258 + 0.965i)T \)
13 \( 1 + (-0.439 - 3.57i)T \)
good2 \( 1 + (-1.60 - 1.60i)T + 2iT^{2} \)
5 \( 1 + (-0.0631 + 0.235i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-4.20 + 4.20i)T - 11iT^{2} \)
17 \( 1 + (0.508 - 0.880i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.70 + 6.34i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-2.71 + 4.69i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.22iT - 29T^{2} \)
31 \( 1 + (-2.31 - 0.619i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (2.18 - 8.14i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-3.35 + 0.898i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-2.25 + 1.30i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.38 - 5.17i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + 7.04iT - 53T^{2} \)
59 \( 1 + (1.90 - 1.90i)T - 59iT^{2} \)
61 \( 1 + (6.13 + 10.6i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.77 + 10.3i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (6.13 - 1.64i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (5.12 + 5.12i)T + 73iT^{2} \)
79 \( 1 + (1.60 - 2.78i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (15.8 - 4.24i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-5.11 - 1.36i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (8.58 + 2.30i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.75186169188742387403183022613, −9.027525071413972449788450460267, −8.422331248806824067006144842435, −7.18395627754937532764662685799, −6.61843841228441676173335732078, −6.22549396110161959621175344709, −5.01263590029987968829307093272, −4.41172932801017744112311952079, −3.13127073888811422437254465007, −1.18330462765131902387873941427, 1.39557783300900553560785056824, 2.79241378910263629946040412342, 3.94747671522802940580252130805, 4.42905481245060270870903312983, 5.53261544467964043337348140846, 6.09820221539757837521172065716, 7.39267086482668097196471383961, 8.852884810322316129098390852074, 9.822366732157324105396269896224, 10.31047580194305585355120098406

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