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Properties

Label 2-2700-15.2-c1-0-17
Degree $2$
Conductor $2700$
Sign $0.229 + 0.973i$
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  + (−0.224 − 0.224i)7-s + 2.78i·11-s + (1.22 − 1.22i)13-s + (2.78 − 2.78i)17-s − 3.89i·19-s + (−6.19 − 6.19i)23-s − 2.78·29-s + 0.449·31-s + (−3.67 − 3.67i)37-s + 9.60i·41-s + (−0.550 + 0.550i)43-s − 6.89i·49-s + (6.19 + 6.19i)53-s + 12.3·59-s + 12.7·61-s + ⋯
L(s)  = 1  + (−0.0849 − 0.0849i)7-s + 0.839i·11-s + (0.339 − 0.339i)13-s + (0.675 − 0.675i)17-s − 0.894i·19-s + (−1.29 − 1.29i)23-s − 0.517·29-s + 0.0807·31-s + (−0.604 − 0.604i)37-s + 1.49i·41-s + (−0.0839 + 0.0839i)43-s − 0.985i·49-s + (0.850 + 0.850i)53-s + 1.61·59-s + 1.63·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2700\)    =    \(2^{2} \cdot 3^{3} \cdot 5^{2}\)
Sign: $0.229 + 0.973i$
Analytic conductor: \(21.5596\)
Root analytic conductor: \(4.64323\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2700} (1457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2700,\ (\ :1/2),\ 0.229 + 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.430547415\)
\(L(\frac12)\) \(\approx\) \(1.430547415\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.224 + 0.224i)T + 7iT^{2} \)
11 \( 1 - 2.78iT - 11T^{2} \)
13 \( 1 + (-1.22 + 1.22i)T - 13iT^{2} \)
17 \( 1 + (-2.78 + 2.78i)T - 17iT^{2} \)
19 \( 1 + 3.89iT - 19T^{2} \)
23 \( 1 + (6.19 + 6.19i)T + 23iT^{2} \)
29 \( 1 + 2.78T + 29T^{2} \)
31 \( 1 - 0.449T + 31T^{2} \)
37 \( 1 + (3.67 + 3.67i)T + 37iT^{2} \)
41 \( 1 - 9.60iT - 41T^{2} \)
43 \( 1 + (0.550 - 0.550i)T - 43iT^{2} \)
47 \( 1 - 47iT^{2} \)
53 \( 1 + (-6.19 - 6.19i)T + 53iT^{2} \)
59 \( 1 - 12.3T + 59T^{2} \)
61 \( 1 - 12.7T + 61T^{2} \)
67 \( 1 + (8.67 + 8.67i)T + 67iT^{2} \)
71 \( 1 + 9.60iT - 71T^{2} \)
73 \( 1 + (-6.22 + 6.22i)T - 73iT^{2} \)
79 \( 1 + 9.24iT - 79T^{2} \)
83 \( 1 + (-3.41 - 3.41i)T + 83iT^{2} \)
89 \( 1 + 12.3T + 89T^{2} \)
97 \( 1 + (-4.77 - 4.77i)T + 97iT^{2} \)
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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

−8.643654749292668139580503277859, −7.913347625806846435231894411417, −7.15169166475003781984934483348, −6.46765554729837300793103539225, −5.54050408324668236643860698902, −4.74001272333540269540831189781, −3.92729397877968008194656015514, −2.88005924042742190778937956188, −1.93741375533437107891505701835, −0.49690290967826565765311311372, 1.20785186160922612047496048353, 2.26655265328863083743758377728, 3.69138234244262438303992168272, 3.83370756238277363106590910615, 5.45854592125704950171962323717, 5.72685450849007484270646832355, 6.67691835963952244572140133625, 7.57494462755273002872411363300, 8.310416405085199627536239390812, 8.835822702993488191626112915543

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