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Properties

Label 2-1755-9.4-c1-0-17
Degree $2$
Conductor $1755$
Sign $-0.953 + 0.301i$
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  + (1.22 + 2.12i)2-s + (−1.99 + 3.45i)4-s + (−0.5 + 0.866i)5-s + (1.96 + 3.40i)7-s − 4.88·8-s − 2.44·10-s + (2.33 + 4.04i)11-s + (0.5 − 0.866i)13-s + (−4.81 + 8.34i)14-s + (−1.98 − 3.43i)16-s + 7.96·17-s − 0.120·19-s + (−1.99 − 3.45i)20-s + (−5.72 + 9.90i)22-s + (0.164 − 0.284i)23-s + ⋯
L(s)  = 1  + (0.865 + 1.49i)2-s + (−0.998 + 1.72i)4-s + (−0.223 + 0.387i)5-s + (0.743 + 1.28i)7-s − 1.72·8-s − 0.774·10-s + (0.704 + 1.22i)11-s + (0.138 − 0.240i)13-s + (−1.28 + 2.22i)14-s + (−0.495 − 0.858i)16-s + 1.93·17-s − 0.0277·19-s + (−0.446 − 0.773i)20-s + (−1.21 + 2.11i)22-s + (0.0342 − 0.0592i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 + 0.301i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.953 + 0.301i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1755\)    =    \(3^{3} \cdot 5 \cdot 13\)
Sign: $-0.953 + 0.301i$
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: \((2,\ 1755,\ (\ :1/2),\ -0.953 + 0.301i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.966790370\)
\(L(\frac12)\) \(\approx\) \(2.966790370\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.22 - 2.12i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (-1.96 - 3.40i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.33 - 4.04i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 7.96T + 17T^{2} \)
19 \( 1 + 0.120T + 19T^{2} \)
23 \( 1 + (-0.164 + 0.284i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.136 + 0.236i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.28 + 5.69i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 7.43T + 37T^{2} \)
41 \( 1 + (-0.0455 + 0.0788i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.89 + 6.75i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.78 + 10.0i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 11.1T + 53T^{2} \)
59 \( 1 + (-2.77 + 4.81i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.63 + 11.4i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.41 - 5.90i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.32T + 71T^{2} \)
73 \( 1 + 7.37T + 73T^{2} \)
79 \( 1 + (-5.95 - 10.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.774 - 1.34i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 3.59T + 89T^{2} \)
97 \( 1 + (2.63 + 4.57i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.500707631814016659922506676843, −8.492353090818482671594901160095, −8.019782416598001846161672360556, −7.20855917959538033523424777643, −6.55250323007669555643840613822, −5.51910043600990650220491649816, −5.23349739645571880180457486989, −4.16916352100785284366017623661, −3.29865584191410972047096615902, −1.92583134277112157537668157612, 1.00497535191798861004010348903, 1.39225728832040138071204219377, 3.13156208278023414876002660118, 3.65440439884598638387869507519, 4.47445199254547220991765763813, 5.21295503356847196033696074555, 6.12495789849096050781403626353, 7.36564713520051324056350560680, 8.201883235165308395166877329567, 9.083635398859048599391598843721

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