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Properties

Label 2-2700-9.4-c1-0-17
Degree $2$
Conductor $2700$
Sign $-0.894 + 0.446i$
Analytic cond. $21.5596$
Root an. cond. $4.64323$
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  + (−1.91 − 3.32i)7-s + (−0.853 − 1.47i)11-s + (1.85 − 3.21i)13-s + 1.70·17-s + 0.292·19-s + (2.91 − 5.05i)23-s + (4.33 + 7.50i)29-s + (−0.146 + 0.253i)31-s − 11.9·37-s + (−3.48 + 6.02i)41-s + (1.85 + 3.21i)43-s + (−0.936 − 1.62i)47-s + (−3.85 + 6.67i)49-s − 11.6·53-s + (5.83 − 10.1i)59-s + ⋯
L(s)  = 1  + (−0.724 − 1.25i)7-s + (−0.257 − 0.445i)11-s + (0.514 − 0.890i)13-s + 0.414·17-s + 0.0671·19-s + (0.608 − 1.05i)23-s + (0.804 + 1.39i)29-s + (−0.0262 + 0.0455i)31-s − 1.96·37-s + (−0.543 + 0.941i)41-s + (0.282 + 0.489i)43-s + (−0.136 − 0.236i)47-s + (−0.550 + 0.953i)49-s − 1.60·53-s + (0.759 − 1.31i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.446i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2700\)    =    \(2^{2} \cdot 3^{3} \cdot 5^{2}\)
Sign: $-0.894 + 0.446i$
Analytic conductor: \(21.5596\)
Root analytic conductor: \(4.64323\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2700} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2700,\ (\ :1/2),\ -0.894 + 0.446i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9404685984\)
\(L(\frac12)\) \(\approx\) \(0.9404685984\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (1.91 + 3.32i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.853 + 1.47i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.85 + 3.21i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 1.70T + 17T^{2} \)
19 \( 1 - 0.292T + 19T^{2} \)
23 \( 1 + (-2.91 + 5.05i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.33 - 7.50i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.146 - 0.253i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 11.9T + 37T^{2} \)
41 \( 1 + (3.48 - 6.02i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.85 - 3.21i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.936 + 1.62i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 11.6T + 53T^{2} \)
59 \( 1 + (-5.83 + 10.1i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.48 + 12.9i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.77 + 8.26i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 15.9T + 71T^{2} \)
73 \( 1 + 8T + 73T^{2} \)
79 \( 1 + (1 + 1.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.91 - 10.2i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 3T + 89T^{2} \)
97 \( 1 + (1.83 + 3.17i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.402900654493627426831936337660, −7.83031144848917146501357987241, −6.83294887621530252965859309229, −6.46028528538230941270937329305, −5.34072302336934388850410287637, −4.60118666152845393341503083422, −3.34406409913096780305001313131, −3.17040527266928132929744045716, −1.39200937117684130761851785446, −0.31172008586023116237790410932, 1.57510454142084890912136261166, 2.58828000135248508157724550928, 3.43330376508048192791627583045, 4.44805834450361901602473810696, 5.46386767001783806788545456237, 5.99460877203071837859834441933, 6.87009076547302041437705171652, 7.56922939691794595197883618432, 8.730886234318462429880028989170, 8.962799179627987796671804839261

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