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Properties

Label 2-1782-9.4-c1-0-32
Degree $2$
Conductor $1782$
Sign $-0.766 + 0.642i$
Analytic cond. $14.2293$
Root an. cond. $3.77217$
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  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (2 − 3.46i)5-s + (1 + 1.73i)7-s + 0.999·8-s − 3.99·10-s + (−0.5 − 0.866i)11-s + (−2 + 3.46i)13-s + (0.999 − 1.73i)14-s + (−0.5 − 0.866i)16-s − 2·17-s + (1.99 + 3.46i)20-s + (−0.499 + 0.866i)22-s + (3 − 5.19i)23-s + (−5.49 − 9.52i)25-s + 3.99·26-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.894 − 1.54i)5-s + (0.377 + 0.654i)7-s + 0.353·8-s − 1.26·10-s + (−0.150 − 0.261i)11-s + (−0.554 + 0.960i)13-s + (0.267 − 0.462i)14-s + (−0.125 − 0.216i)16-s − 0.485·17-s + (0.447 + 0.774i)20-s + (−0.106 + 0.184i)22-s + (0.625 − 1.08i)23-s + (−1.09 − 1.90i)25-s + 0.784·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1782 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1782 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1782\)    =    \(2 \cdot 3^{4} \cdot 11\)
Sign: $-0.766 + 0.642i$
Analytic conductor: \(14.2293\)
Root analytic conductor: \(3.77217\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1782} (595, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1782,\ (\ :1/2),\ -0.766 + 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.375958269\)
\(L(\frac12)\) \(\approx\) \(1.375958269\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 \)
11 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (-2 + 3.46i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1 - 1.73i)T + (-3.5 + 6.06i)T^{2} \)
13 \( 1 + (2 - 3.46i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (5 + 8.66i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4 + 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + (1 - 1.73i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2 + 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1 - 1.73i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 4T + 53T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4 - 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6 + 10.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 + (5 + 8.66i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2 + 3.46i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 + (-1 - 1.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.984083959703027102146620674907, −8.580013124211980029449715718335, −7.72781542346441114979631301965, −6.44342257669605424274672700874, −5.58951479845520128709309248326, −4.77340038470765418958071093915, −4.16243897164867286248916409755, −2.42231758244668237792097955814, −1.91585407218004497353075962837, −0.57397514602922536623797448722, 1.48230954439127501971352936383, 2.68959847971937645275488320140, 3.59388753817354602304149762447, 5.04095446506136902275559668978, 5.60762133898102671224509621676, 6.72868231985871113681500698886, 7.09965155020083103847013845642, 7.74086374052182286630347156246, 8.819916792474218801125631623247, 9.711885256932617304222197803049

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