Properties

Label 2-1755-9.4-c1-0-44
Degree $2$
Conductor $1755$
Sign $-0.173 - 0.984i$
Analytic cond. $14.0137$
Root an. cond. $3.74349$
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 + (0.500 − 0.866i)4-s + (−0.5 + 0.866i)5-s + (−1 − 1.73i)7-s − 3·8-s + 0.999·10-s + (−0.5 − 0.866i)11-s + (0.5 − 0.866i)13-s + (−0.999 + 1.73i)14-s + (0.500 + 0.866i)16-s − 2·17-s − 3·19-s + (0.499 + 0.866i)20-s + (−0.499 + 0.866i)22-s + (−0.499 − 0.866i)25-s − 0.999·26-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.250 − 0.433i)4-s + (−0.223 + 0.387i)5-s + (−0.377 − 0.654i)7-s − 1.06·8-s + 0.316·10-s + (−0.150 − 0.261i)11-s + (0.138 − 0.240i)13-s + (−0.267 + 0.462i)14-s + (0.125 + 0.216i)16-s − 0.485·17-s − 0.688·19-s + (0.111 + 0.193i)20-s + (−0.106 + 0.184i)22-s + (−0.0999 − 0.173i)25-s − 0.196·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1755\)    =    \(3^{3} \cdot 5 \cdot 13\)
Sign: $-0.173 - 0.984i$
Analytic conductor: \(14.0137\)
Root analytic conductor: \(3.74349\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1755} (1171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 1755,\ (\ :1/2),\ -0.173 - 0.984i)\)

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)$
bad3 \( 1 \)
5 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (0.5 + 0.866i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (1 + 1.73i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 + 3T + 19T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.5 - 4.33i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 5T + 37T^{2} \)
41 \( 1 + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4 - 6.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1 + 1.73i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 14T + 53T^{2} \)
59 \( 1 + (4.5 - 7.79i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7 - 12.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 6T + 73T^{2} \)
79 \( 1 + (-6 - 10.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5 + 8.66i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 2T + 89T^{2} \)
97 \( 1 + (0.5 + 0.866i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.894099906779463724199322950964, −8.065496853768385230930604172136, −7.00117035277627246441829788683, −6.46163334418971755662590960971, −5.58341290204056281625952393212, −4.42325939299105524856234472394, −3.37463358066403399753786183922, −2.57977611524376115471689126586, −1.32547714595512309609106092183, 0, 2.03028915539389220282682487604, 3.03091185132671330206734504038, 4.07926188039614167590725058724, 5.09667498073455144308057212363, 6.17705086673578182753206858978, 6.62093839365958432014770772621, 7.63916520216298262106241334035, 8.240372061130828114126407554162, 9.008135041586029053512543974070

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