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Properties

Label 2-2736-57.50-c1-0-37
Degree $2$
Conductor $2736$
Sign $-0.998 + 0.0456i$
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  + (−0.501 + 0.289i)5-s + 0.198·7-s − 2.48i·11-s + (−4.34 − 2.50i)13-s + (4.01 − 2.31i)17-s + (3.16 − 2.99i)19-s + (−5.14 − 2.97i)23-s + (−2.33 + 4.03i)25-s + (−4.51 + 7.82i)29-s + 1.41i·31-s + (−0.0995 + 0.0574i)35-s + 11.6i·37-s + (−3.60 − 6.25i)41-s + (4.66 + 8.07i)43-s + (1.86 + 1.07i)47-s + ⋯
L(s)  = 1  + (−0.224 + 0.129i)5-s + 0.0750·7-s − 0.749i·11-s + (−1.20 − 0.695i)13-s + (0.973 − 0.562i)17-s + (0.725 − 0.688i)19-s + (−1.07 − 0.619i)23-s + (−0.466 + 0.807i)25-s + (−0.839 + 1.45i)29-s + 0.253i·31-s + (−0.0168 + 0.00971i)35-s + 1.91i·37-s + (−0.563 − 0.976i)41-s + (0.710 + 1.23i)43-s + (0.271 + 0.156i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0456i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0456i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.998 + 0.0456i$
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: \((2,\ 2736,\ (\ :1/2),\ -0.998 + 0.0456i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2158704923\)
\(L(\frac12)\) \(\approx\) \(0.2158704923\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.16 + 2.99i)T \)
good5 \( 1 + (0.501 - 0.289i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 - 0.198T + 7T^{2} \)
11 \( 1 + 2.48iT - 11T^{2} \)
13 \( 1 + (4.34 + 2.50i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-4.01 + 2.31i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (5.14 + 2.97i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.51 - 7.82i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.41iT - 31T^{2} \)
37 \( 1 - 11.6iT - 37T^{2} \)
41 \( 1 + (3.60 + 6.25i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.66 - 8.07i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.86 - 1.07i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.62 - 2.81i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.59 + 7.95i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.46 - 4.27i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.17 + 1.83i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (8.07 + 13.9i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.02 + 8.70i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.26 + 0.730i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 0.592iT - 83T^{2} \)
89 \( 1 + (2.78 - 4.82i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (9.32 - 5.38i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.323481476443051665888399900546, −7.65896275113110449422942031587, −7.12833592954211002152002780646, −6.07480448349941770358717646012, −5.28570038503859340745907279872, −4.66779985377645239989621354179, −3.31717201398530911579496030101, −2.93138826143343299438593259472, −1.46019637170508103917449606704, −0.06763869677098977988408381267, 1.64474475890879295284449794728, 2.47058313178154359171596442422, 3.85137354222669968495970015370, 4.29387262633713087955398581912, 5.44337260850511429704505021458, 5.97726792224103127814002142370, 7.16793933633454488271400222973, 7.65120448787389551368295789838, 8.246364460941784405809738176634, 9.494399430945947150364816343469

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