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Properties

Label 2-1755-9.4-c1-0-25
Degree $2$
Conductor $1755$
Sign $0.696 - 0.717i$
Analytic cond. $14.0137$
Root an. cond. $3.74349$
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  + (0.817 + 1.41i)2-s + (−0.335 + 0.580i)4-s + (−0.5 + 0.866i)5-s + (−1.06 − 1.84i)7-s + 2.17·8-s − 1.63·10-s + (−0.0263 − 0.0455i)11-s + (0.5 − 0.866i)13-s + (1.74 − 3.01i)14-s + (2.44 + 4.23i)16-s + 2.48·17-s + 2.13·19-s + (−0.335 − 0.580i)20-s + (0.0430 − 0.0744i)22-s + (2.46 − 4.27i)23-s + ⋯
L(s)  = 1  + (0.577 + 1.00i)2-s + (−0.167 + 0.290i)4-s + (−0.223 + 0.387i)5-s + (−0.402 − 0.697i)7-s + 0.768·8-s − 0.516·10-s + (−0.00793 − 0.0137i)11-s + (0.138 − 0.240i)13-s + (0.465 − 0.805i)14-s + (0.611 + 1.05i)16-s + 0.601·17-s + 0.488·19-s + (−0.0749 − 0.129i)20-s + (0.00917 − 0.0158i)22-s + (0.514 − 0.890i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.696 - 0.717i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.696 - 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1755\)    =    \(3^{3} \cdot 5 \cdot 13\)
Sign: $0.696 - 0.717i$
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: \(0\)
Selberg data: \((2,\ 1755,\ (\ :1/2),\ 0.696 - 0.717i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.508110102\)
\(L(\frac12)\) \(\approx\) \(2.508110102\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.817 - 1.41i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (1.06 + 1.84i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.0263 + 0.0455i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 2.48T + 17T^{2} \)
19 \( 1 - 2.13T + 19T^{2} \)
23 \( 1 + (-2.46 + 4.27i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.24 + 2.15i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.08 + 7.06i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 1.11T + 37T^{2} \)
41 \( 1 + (2.73 - 4.74i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.73 - 8.20i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.88 - 8.45i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 3.64T + 53T^{2} \)
59 \( 1 + (-3.74 + 6.48i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.89 - 5.00i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.11 + 5.40i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.50T + 71T^{2} \)
73 \( 1 + 1.10T + 73T^{2} \)
79 \( 1 + (-7.80 - 13.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.244 - 0.423i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 4.64T + 89T^{2} \)
97 \( 1 + (3.67 + 6.35i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.470703340830924850887643031934, −8.160228211060673021777442010483, −7.68884561010927449244879747218, −6.88430583494341213937724763310, −6.28855228326369253098411432276, −5.49864483093671949058340628068, −4.51649758141977871816073781795, −3.79125616930121155006084331905, −2.66849411578718148533836733064, −0.974457025995900285741903373232, 1.15621084888838979330073621512, 2.30023466744546248377489215786, 3.27804740156655574015080342206, 3.90679658278460842449373928491, 5.07872792313851868842403988176, 5.56899946095473501407496018985, 6.88803674691948296951147497171, 7.58159690550773791788094418506, 8.630284722784755518174742570058, 9.240352746876257821332491179866

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