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Properties

Label 2-819-117.43-c1-0-33
Degree $2$
Conductor $819$
Sign $-0.525 - 0.850i$
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  + (−1.33 + 0.773i)2-s + (−0.391 + 1.68i)3-s + (0.196 − 0.339i)4-s + (0.780 − 0.450i)5-s + (−0.779 − 2.56i)6-s + i·7-s − 2.48i·8-s + (−2.69 − 1.32i)9-s + (−0.697 + 1.20i)10-s + (3.38 − 1.95i)11-s + (0.496 + 0.464i)12-s + (3.57 − 0.445i)13-s + (−0.773 − 1.33i)14-s + (0.454 + 1.49i)15-s + (2.31 + 4.01i)16-s + (2.34 + 4.05i)17-s + ⋯
L(s)  = 1  + (−0.947 + 0.546i)2-s + (−0.226 + 0.974i)3-s + (0.0980 − 0.169i)4-s + (0.349 − 0.201i)5-s + (−0.318 − 1.04i)6-s + 0.377i·7-s − 0.879i·8-s + (−0.897 − 0.440i)9-s + (−0.220 + 0.381i)10-s + (1.02 − 0.589i)11-s + (0.143 + 0.133i)12-s + (0.992 − 0.123i)13-s + (−0.206 − 0.357i)14-s + (0.117 + 0.385i)15-s + (0.578 + 1.00i)16-s + (0.567 + 0.983i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.525 - 0.850i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.525 - 0.850i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.446748 + 0.800716i\)
\(L(\frac12)\) \(\approx\) \(0.446748 + 0.800716i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.391 - 1.68i)T \)
7 \( 1 - iT \)
13 \( 1 + (-3.57 + 0.445i)T \)
good2 \( 1 + (1.33 - 0.773i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (-0.780 + 0.450i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.38 + 1.95i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.34 - 4.05i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.626 - 0.361i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 3.05T + 23T^{2} \)
29 \( 1 + (-4.05 - 7.02i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.726 - 0.419i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.71 + 2.72i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 12.4iT - 41T^{2} \)
43 \( 1 - 1.14T + 43T^{2} \)
47 \( 1 + (-10.6 - 6.15i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 1.75T + 53T^{2} \)
59 \( 1 + (2.96 + 1.71i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + 3.28T + 61T^{2} \)
67 \( 1 - 4.00iT - 67T^{2} \)
71 \( 1 + (2.55 - 1.47i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 2.98iT - 73T^{2} \)
79 \( 1 + (5.91 - 10.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (14.2 + 8.23i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-8.81 - 5.08i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 11.7iT - 97T^{2} \)
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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

−10.43522752978860848279462577185, −9.357474297128213618360408654701, −8.849842276092423610599597033546, −8.450434126158389372993944179414, −7.12258229470190036925690388878, −6.10697492105122874010089408023, −5.52131914243489901667703176388, −4.03099355762417437066978904227, −3.37687146905590521121349677254, −1.20057886955871178649627685712, 0.811619222205998555425575664147, 1.75032599521691933653844124035, 2.91235144461779365915386558644, 4.54418899718269110789872549600, 5.80780494040636457743645053359, 6.58904864737333692813653785127, 7.49267149831365175665269266678, 8.371805811783538677957486058070, 9.149598094166395627367657315167, 9.959516352579117457345685694805

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