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Properties

Label 2-1782-99.32-c1-0-17
Degree $2$
Conductor $1782$
Sign $0.254 + 0.966i$
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 + (−1.22 − 0.707i)5-s + (−3.67 + 2.12i)7-s + 0.999·8-s + 1.41i·10-s + (0.275 + 3.30i)11-s + (−3.67 − 2.12i)13-s + (3.67 + 2.12i)14-s + (−0.5 − 0.866i)16-s + (1.22 − 0.707i)20-s + (2.72 − 1.89i)22-s + (−1.22 − 0.707i)23-s + (−1.50 − 2.59i)25-s + 4.24i·26-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.547 − 0.316i)5-s + (−1.38 + 0.801i)7-s + 0.353·8-s + 0.447i·10-s + (0.0829 + 0.996i)11-s + (−1.01 − 0.588i)13-s + (0.981 + 0.566i)14-s + (−0.125 − 0.216i)16-s + (0.273 − 0.158i)20-s + (0.580 − 0.403i)22-s + (−0.255 − 0.147i)23-s + (−0.300 − 0.519i)25-s + 0.832i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1782 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.254 + 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1782 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.254 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1782\)    =    \(2 \cdot 3^{4} \cdot 11\)
Sign: $0.254 + 0.966i$
Analytic conductor: \(14.2293\)
Root analytic conductor: \(3.77217\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1782} (1187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1782,\ (\ :1/2),\ 0.254 + 0.966i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6544409410\)
\(L(\frac12)\) \(\approx\) \(0.6544409410\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.275 - 3.30i)T \)
good5 \( 1 + (1.22 + 0.707i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (3.67 - 2.12i)T + (3.5 - 6.06i)T^{2} \)
13 \( 1 + (3.67 + 2.12i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + (1.22 + 0.707i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (3 - 5.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-7.34 + 4.24i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-8.57 + 4.94i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 7.07iT - 53T^{2} \)
59 \( 1 + (-9.79 - 5.65i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.67 - 2.12i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.07iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + (3.67 - 2.12i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (6 + 10.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 5.65iT - 89T^{2} \)
97 \( 1 + (4 + 6.92i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.211925270909897509975970063678, −8.540571578362298099429860486946, −7.60759304905429591856607694562, −6.90064369407422780882562963141, −5.89201982474065981155710445429, −4.87521099930198465333437961461, −3.98075898743167482920002944655, −2.94187678824421161097350611273, −2.21247557611445680049458750101, −0.42188120984316742534202125269, 0.74363159552160212331470135730, 2.66026681686028674561785998861, 3.66187874920585036620777237553, 4.40316903775069502065927462582, 5.69708361653108591219008397802, 6.41842350529911858149355802251, 7.14422208878964793069959972644, 7.65834377029598797004971051112, 8.651314861657736801677947857926, 9.478591576070531517002376744597

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