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Properties

Label 2-2736-57.50-c1-0-30
Degree $2$
Conductor $2736$
Sign $0.945 + 0.326i$
Analytic cond. $21.8470$
Root an. cond. $4.67408$
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  + (1.80 − 1.04i)5-s + 4.37·7-s − 5.52i·11-s + (4.60 + 2.65i)13-s + (−0.666 + 0.384i)17-s + (2.87 + 3.27i)19-s + (5.78 + 3.33i)23-s + (−0.331 + 0.574i)25-s + (−1.50 + 2.60i)29-s − 2.12i·31-s + (7.88 − 4.55i)35-s + 8.39i·37-s + (−2.76 − 4.78i)41-s + (−5.71 − 9.90i)43-s + (3.03 + 1.75i)47-s + ⋯
L(s)  = 1  + (0.806 − 0.465i)5-s + 1.65·7-s − 1.66i·11-s + (1.27 + 0.737i)13-s + (−0.161 + 0.0933i)17-s + (0.659 + 0.751i)19-s + (1.20 + 0.696i)23-s + (−0.0663 + 0.114i)25-s + (−0.279 + 0.484i)29-s − 0.381i·31-s + (1.33 − 0.769i)35-s + 1.37i·37-s + (−0.431 − 0.747i)41-s + (−0.872 − 1.51i)43-s + (0.442 + 0.255i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.326i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.945 + 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.945 + 0.326i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ 0.945 + 0.326i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.927875691\)
\(L(\frac12)\) \(\approx\) \(2.927875691\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19 \( 1 + (-2.87 - 3.27i)T \)
good5 \( 1 + (-1.80 + 1.04i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 - 4.37T + 7T^{2} \)
11 \( 1 + 5.52iT - 11T^{2} \)
13 \( 1 + (-4.60 - 2.65i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.666 - 0.384i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (-5.78 - 3.33i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.50 - 2.60i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 2.12iT - 31T^{2} \)
37 \( 1 - 8.39iT - 37T^{2} \)
41 \( 1 + (2.76 + 4.78i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.71 + 9.90i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.03 - 1.75i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.59 - 6.23i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.46 + 2.53i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.90 - 6.76i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (11.5 + 6.65i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.04 - 1.81i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.258 + 0.447i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.23 - 1.87i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 12.0iT - 83T^{2} \)
89 \( 1 + (6.22 - 10.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.74 - 2.74i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.787202894326736455682601508522, −8.235378738598359261483410095640, −7.39319886598944908044850534719, −6.29177987063355820737931449402, −5.56841668436896148178994906745, −5.12647877382053024537630745532, −4.01972953160022429086372843001, −3.13663362966575544508200996016, −1.60350163965690337342909370997, −1.27902090197783093060979161568, 1.25703370345617512698893860805, 2.04597308024674678416448370003, 2.99273032530745606147127443661, 4.36587680980866006071502127722, 4.92073813572168114311255910344, 5.69891754047840866813149433526, 6.65398215830078902176184424853, 7.35736735102422097714981800250, 8.088038392902411180857644186455, 8.830338941445781535775338519606

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