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Properties

Label 2-2700-180.83-c0-0-1
Degree $2$
Conductor $2700$
Sign $0.395 - 0.918i$
Analytic cond. $1.34747$
Root an. cond. $1.16080$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.258 + 0.965i)2-s + (−0.866 + 0.499i)4-s + (1.67 − 0.448i)7-s + (−0.707 − 0.707i)8-s + (0.866 + 1.50i)14-s + (0.500 − 0.866i)16-s + (−0.258 + 0.965i)23-s + (−1.22 + 1.22i)28-s + (0.866 − 1.5i)29-s + (0.965 + 0.258i)32-s + (1.5 − 0.866i)41-s − 46-s + (0.258 + 0.965i)47-s + (1.73 − 1.00i)49-s + (−1.5 − 0.866i)56-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)2-s + (−0.866 + 0.499i)4-s + (1.67 − 0.448i)7-s + (−0.707 − 0.707i)8-s + (0.866 + 1.50i)14-s + (0.500 − 0.866i)16-s + (−0.258 + 0.965i)23-s + (−1.22 + 1.22i)28-s + (0.866 − 1.5i)29-s + (0.965 + 0.258i)32-s + (1.5 − 0.866i)41-s − 46-s + (0.258 + 0.965i)47-s + (1.73 − 1.00i)49-s + (−1.5 − 0.866i)56-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.395 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.395 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2700\)    =    \(2^{2} \cdot 3^{3} \cdot 5^{2}\)
Sign: $0.395 - 0.918i$
Analytic conductor: \(1.34747\)
Root analytic conductor: \(1.16080\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2700} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2700,\ (\ :0),\ 0.395 - 0.918i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.519590665\)
\(L(\frac12)\) \(\approx\) \(1.519590665\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 - 0.965i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-1.67 + 0.448i)T + (0.866 - 0.5i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.866 - 0.5i)T^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \)
29 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.866 + 0.5i)T^{2} \)
47 \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.448 - 1.67i)T + (-0.866 - 0.5i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \)
89 \( 1 + 1.73T + T^{2} \)
97 \( 1 + (-0.866 + 0.5i)T^{2} \)
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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

−8.892015014007622288782517267258, −8.152032612270361228074895036066, −7.67365525204566255830689872342, −7.06211438150438996267739018992, −5.95271093355942766364428204855, −5.39379738208967722326496971503, −4.40601836671386213765639400548, −4.07615305001152466489610069950, −2.63007653405636122465019883807, −1.21944424120963385988471203661, 1.23617580004521859076255459043, 2.12808453475780016851709470109, 3.02103935700100006035127423170, 4.23457296693519691764216003848, 4.81488335838217564855373210724, 5.46427136134700145990838466243, 6.40899724280893882907453534729, 7.61028843606626701805405724920, 8.395246678830280283029750121488, 8.811051653345835433168342153390

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