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Properties

Label 2-1782-9.4-c1-0-37
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (−1 − 1.73i)7-s − 0.999·8-s + (0.5 + 0.866i)11-s + (2 − 3.46i)13-s + (0.999 − 1.73i)14-s + (−0.5 − 0.866i)16-s − 6·17-s − 4·19-s + (−0.499 + 0.866i)22-s + (−3 + 5.19i)23-s + (2.5 + 4.33i)25-s + 3.99·26-s + 1.99·28-s + (−3 − 5.19i)29-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.377 − 0.654i)7-s − 0.353·8-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 − 1.45·17-s − 0.917·19-s + (−0.106 + 0.184i)22-s + (−0.625 + 1.08i)23-s + (0.5 + 0.866i)25-s + 0.784·26-s + 0.377·28-s + (−0.557 − 0.964i)29-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: \(1\)
Selberg data: \((2,\ 1782,\ (\ :1/2),\ -0.766 + 0.642i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 - 4.33i)T^{2} \)
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 \( 1 + 6T + 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4 - 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 10T + 37T^{2} \)
41 \( 1 + (3 - 5.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4 + 6.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 + (7 + 12.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6 - 10.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + (7 + 12.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.823522586597613641969185713155, −8.156049874949185859726919672775, −7.17818355183004524189067535742, −6.70364137603408796901837619544, −5.78212363186818498779572422052, −4.94699855274743047318720614922, −3.93932028210727303210167668542, −3.30630139689748606690655339026, −1.81584022054677731782935524934, 0, 1.84607312053401710187327853531, 2.56830513159392114060902556043, 3.83457998402915164076033113796, 4.42693962824454371013937712665, 5.52010695050081761890447216609, 6.42119987478575249054089169467, 6.86435930265616229656004053785, 8.465127065966630089161874736856, 8.796616134912811635432916400572

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