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Properties

Label 2-2700-9.7-c1-0-10
Degree $2$
Conductor $2700$
Sign $0.941 + 0.335i$
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  + (1.88 − 3.26i)7-s + (−1.77 + 3.07i)11-s + (−0.501 − 0.869i)13-s − 1.56·17-s + 7.21·19-s + (3.09 + 5.35i)23-s + (−1.5 + 2.59i)29-s + (2.89 + 5.00i)31-s + 0.851·37-s + (−1.55 − 2.70i)41-s + (1.35 − 2.34i)43-s + (4.87 − 8.44i)47-s + (−3.60 − 6.24i)49-s + 11.2·53-s + (−4.83 − 8.36i)59-s + ⋯
L(s)  = 1  + (0.712 − 1.23i)7-s + (−0.534 + 0.925i)11-s + (−0.139 − 0.241i)13-s − 0.378·17-s + 1.65·19-s + (0.644 + 1.11i)23-s + (−0.278 + 0.482i)29-s + (0.519 + 0.899i)31-s + 0.139·37-s + (−0.243 − 0.421i)41-s + (0.206 − 0.357i)43-s + (0.710 − 1.23i)47-s + (−0.515 − 0.892i)49-s + 1.54·53-s + (−0.629 − 1.08i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.941 + 0.335i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.941 + 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2700\)    =    \(2^{2} \cdot 3^{3} \cdot 5^{2}\)
Sign: $0.941 + 0.335i$
Analytic conductor: \(21.5596\)
Root analytic conductor: \(4.64323\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2700} (1801, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2700,\ (\ :1/2),\ 0.941 + 0.335i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.986403149\)
\(L(\frac12)\) \(\approx\) \(1.986403149\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.88 + 3.26i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.77 - 3.07i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.501 + 0.869i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 1.56T + 17T^{2} \)
19 \( 1 - 7.21T + 19T^{2} \)
23 \( 1 + (-3.09 - 5.35i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.89 - 5.00i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 0.851T + 37T^{2} \)
41 \( 1 + (1.55 + 2.70i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.35 + 2.34i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.87 + 8.44i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 11.2T + 53T^{2} \)
59 \( 1 + (4.83 + 8.36i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.10 + 7.11i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.45 - 2.52i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 7.21T + 71T^{2} \)
73 \( 1 - 16.7T + 73T^{2} \)
79 \( 1 + (5.49 - 9.52i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.09 + 8.82i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 5.33T + 89T^{2} \)
97 \( 1 + (-1.93 + 3.34i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.771430590943824754610095974491, −7.81597775896926414331592619076, −7.30919718728323365754068831719, −6.85587565404956699641770173386, −5.36224042208848701953241061444, −5.03226009117285559659216267086, −4.02491170338667822204277970780, −3.19000608748815382774050145517, −1.91385283016010955266811392888, −0.875242158820661463850657713978, 0.947700168423578997461150602929, 2.37433364500420721669930523325, 2.91952223428179444042190126113, 4.21261655600483082048025218352, 5.14108890136113324362531240167, 5.67866967297742533706291599638, 6.45862429720658218315355467484, 7.57278937461128077801077187913, 8.125919388889243208193785376251, 8.918448191701633499946752094735

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