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Properties

Label 2-2700-9.4-c1-0-13
Degree $2$
Conductor $2700$
Sign $0.870 + 0.492i$
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  + (2.05 + 3.55i)7-s + (−1.90 − 3.30i)11-s + (2.90 − 5.03i)13-s + 3.81·17-s − 1.81·19-s + (−1.05 + 1.81i)23-s + (−3.60 − 6.23i)29-s + (0.908 − 1.57i)31-s + 6.01·37-s + (5.50 − 9.54i)41-s + (2.90 + 5.03i)43-s + (−5.95 − 10.3i)47-s + (−4.90 + 8.50i)49-s + 4.20·53-s + (−2.10 + 3.63i)59-s + ⋯
L(s)  = 1  + (0.774 + 1.34i)7-s + (−0.575 − 0.996i)11-s + (0.806 − 1.39i)13-s + 0.925·17-s − 0.416·19-s + (−0.219 + 0.379i)23-s + (−0.668 − 1.15i)29-s + (0.163 − 0.282i)31-s + 0.989·37-s + (0.860 − 1.49i)41-s + (0.443 + 0.768i)43-s + (−0.869 − 1.50i)47-s + (−0.701 + 1.21i)49-s + 0.577·53-s + (−0.273 + 0.473i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.870 + 0.492i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.870 + 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2700\)    =    \(2^{2} \cdot 3^{3} \cdot 5^{2}\)
Sign: $0.870 + 0.492i$
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: \((2,\ 2700,\ (\ :1/2),\ 0.870 + 0.492i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.961237410\)
\(L(\frac12)\) \(\approx\) \(1.961237410\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.05 - 3.55i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.90 + 3.30i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.90 + 5.03i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 3.81T + 17T^{2} \)
19 \( 1 + 1.81T + 19T^{2} \)
23 \( 1 + (1.05 - 1.81i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.60 + 6.23i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.908 + 1.57i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 6.01T + 37T^{2} \)
41 \( 1 + (-5.50 + 9.54i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.90 - 5.03i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.95 + 10.3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 4.20T + 53T^{2} \)
59 \( 1 + (2.10 - 3.63i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.50 - 2.61i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.85 + 3.21i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.01T + 71T^{2} \)
73 \( 1 + 8T + 73T^{2} \)
79 \( 1 + (1 + 1.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.94 - 3.37i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 3T + 89T^{2} \)
97 \( 1 + (-6.10 - 10.5i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.589761240929105085185838217709, −8.094880856036119935017263043591, −7.60293010147385441110510703474, −6.04335509777119053219477982486, −5.76217585547373646032717990970, −5.15660285007684722565239450521, −3.88140398122170733208044523213, −2.97418453764095580050426577866, −2.13820368815282990829090580382, −0.73997374928975439422391806316, 1.14228677956565121942267908361, 1.97303616622529500376890638882, 3.35328980435639262152663000727, 4.40400062566553948362490460495, 4.62102914130976964138630798189, 5.86844958813734350532571810431, 6.77464528671963280945970548366, 7.43354777361671005951964196513, 7.978359864583863095768875350146, 8.874289716000017091686042749068

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