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Properties

Label 2-819-91.4-c1-0-3
Degree $2$
Conductor $819$
Sign $-0.183 - 0.983i$
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  − 1.46i·2-s − 0.133·4-s + (−2.51 + 1.45i)5-s + (1.03 + 2.43i)7-s − 2.72i·8-s + (2.12 + 3.67i)10-s + (−0.0497 + 0.0287i)11-s + (−2.30 + 2.77i)13-s + (3.55 − 1.51i)14-s − 4.24·16-s − 4.29·17-s + (−7.03 − 4.06i)19-s + (0.335 − 0.193i)20-s + (0.0419 + 0.0726i)22-s − 7.31·23-s + ⋯
L(s)  = 1  − 1.03i·2-s − 0.0667·4-s + (−1.12 + 0.649i)5-s + (0.392 + 0.919i)7-s − 0.963i·8-s + (0.671 + 1.16i)10-s + (−0.0150 + 0.00866i)11-s + (−0.638 + 0.769i)13-s + (0.949 − 0.405i)14-s − 1.06·16-s − 1.04·17-s + (−1.61 − 0.932i)19-s + (0.0750 − 0.0433i)20-s + (0.00894 + 0.0154i)22-s − 1.52·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.183 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.183 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.183 - 0.983i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (550, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.183 - 0.983i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.229318 + 0.276079i\)
\(L(\frac12)\) \(\approx\) \(0.229318 + 0.276079i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.03 - 2.43i)T \)
13 \( 1 + (2.30 - 2.77i)T \)
good2 \( 1 + 1.46iT - 2T^{2} \)
5 \( 1 + (2.51 - 1.45i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.0497 - 0.0287i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + 4.29T + 17T^{2} \)
19 \( 1 + (7.03 + 4.06i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 7.31T + 23T^{2} \)
29 \( 1 + (0.698 - 1.20i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.53 - 3.19i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.87iT - 37T^{2} \)
41 \( 1 + (-5.28 - 3.05i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.07 + 1.87i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.86 - 2.80i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.02 + 10.4i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 14.2iT - 59T^{2} \)
61 \( 1 + (-3.41 + 5.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.42 + 4.86i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.32 - 0.763i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.94 + 3.43i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.34 + 2.32i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 11.9iT - 83T^{2} \)
89 \( 1 + 1.26iT - 89T^{2} \)
97 \( 1 + (7.43 - 4.28i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.75282586210672457878359942050, −9.844635134049185496623411389531, −8.844046119817581635330957414265, −8.071090401284342142518374997346, −6.91753424333110287628702131742, −6.37930129287963556501764543938, −4.66808688689862812099602109702, −3.99242440684449153759996012798, −2.72216658689593289454875863596, −2.04887175234868014469165982344, 0.16075916120644197105127492431, 2.19073450315291055262048987327, 4.04600832277189505422069596058, 4.50194763746556287654135049218, 5.71670838544520256037642456205, 6.64434008869947933215412096979, 7.60972959318460171082886557232, 8.064616890964673371104818443560, 8.615270088215482734373829952299, 10.05234446857130668702636158696

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