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Properties

Label 2-1782-9.4-c1-0-25
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 + (1 − 1.73i)5-s + (2 + 3.46i)7-s − 0.999·8-s + 1.99·10-s + (−0.5 − 0.866i)11-s + (3 − 5.19i)13-s + (−1.99 + 3.46i)14-s + (−0.5 − 0.866i)16-s − 2·17-s + 4·19-s + (0.999 + 1.73i)20-s + (0.499 − 0.866i)22-s + (2 − 3.46i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.447 − 0.774i)5-s + (0.755 + 1.30i)7-s − 0.353·8-s + 0.632·10-s + (−0.150 − 0.261i)11-s + (0.832 − 1.44i)13-s + (−0.534 + 0.925i)14-s + (−0.125 − 0.216i)16-s − 0.485·17-s + 0.917·19-s + (0.223 + 0.387i)20-s + (0.106 − 0.184i)22-s + (0.417 − 0.722i)23-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

\(L(1)\) \(\approx\) \(2.533481929\)
\(L(\frac12)\) \(\approx\) \(2.533481929\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-2 - 3.46i)T + (-3.5 + 6.06i)T^{2} \)
13 \( 1 + (-3 + 5.19i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 + (3 - 5.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2 + 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (6 + 10.3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 + (-6 + 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7 - 12.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2 - 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2 - 3.46i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 + (-7 - 12.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.034384297390905400548348403705, −8.436669977108433953724798262909, −8.121163204432596674383784298862, −6.86869760994811571956819844769, −5.91866392055054951106521898505, −5.23895898386197187467614266012, −4.95941123889200806256257287817, −3.48279463493006215012458586322, −2.50452863707612597124360953207, −1.10490908248658394564698272641, 1.15127893624843364133223562662, 2.09958734761892235686198230314, 3.27992994922626912873065124973, 4.21339163362861451125180654096, 4.79195880012068590063256394188, 6.04873126165091004815406788179, 6.77560937620792065712276556211, 7.48494936392234389853953846666, 8.451011071456022712355878435702, 9.540772325836092351020974804585

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