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Properties

Label 2-2700-5.2-c0-0-1
Degree $2$
Conductor $2700$
Sign $0.991 + 0.130i$
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  + (1.22 − 1.22i)13-s + 2i·19-s + 31-s + (−1.22 − 1.22i)37-s + (1.22 − 1.22i)43-s − i·49-s + 2·61-s + (1.22 + 1.22i)67-s + (−1.22 + 1.22i)73-s + i·79-s + (1.22 − 1.22i)103-s + i·109-s + ⋯
L(s)  = 1  + (1.22 − 1.22i)13-s + 2i·19-s + 31-s + (−1.22 − 1.22i)37-s + (1.22 − 1.22i)43-s − i·49-s + 2·61-s + (1.22 + 1.22i)67-s + (−1.22 + 1.22i)73-s + i·79-s + (1.22 − 1.22i)103-s + i·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2700\)    =    \(2^{2} \cdot 3^{3} \cdot 5^{2}\)
Sign: $0.991 + 0.130i$
Analytic conductor: \(1.34747\)
Root analytic conductor: \(1.16080\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2700} (757, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2700,\ (\ :0),\ 0.991 + 0.130i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.297965813\)
\(L(\frac12)\) \(\approx\) \(1.297965813\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (-1.22 + 1.22i)T - iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 - 2iT - T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T + T^{2} \)
37 \( 1 + (1.22 + 1.22i)T + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (-1.22 + 1.22i)T - iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 2T + T^{2} \)
67 \( 1 + (-1.22 - 1.22i)T + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (1.22 - 1.22i)T - iT^{2} \)
79 \( 1 - iT - T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + iT^{2} \)
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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

−8.735977983437618039404429573394, −8.362840864048794837406988551000, −7.59441891993999056083147020372, −6.70169838625221038757568006058, −5.71740550635878477451481351510, −5.45043475070020935741113170884, −3.97102124304183498248020095326, −3.55684741305562816054161441297, −2.31955420374819433728691995132, −1.09265288240085115016260609114, 1.17284980262875254234453558967, 2.39624914230476532229902095537, 3.38036879735839884552468339775, 4.39341604966668945628835890997, 4.97837272334095820451592286171, 6.21420254874135949989538643019, 6.62433744199030889679757523491, 7.46444893417887150138404749193, 8.468310297994447927864039008316, 8.970387658313361735059413309753

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