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Properties

Label 2-2736-57.50-c1-0-23
Degree $2$
Conductor $2736$
Sign $0.985 - 0.169i$
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  + (2.00 − 1.15i)5-s + 0.442·7-s + 4.35i·11-s + (1.90 + 1.09i)13-s + (1.76 − 1.02i)17-s + (2.76 − 3.36i)19-s + (−0.0969 − 0.0559i)23-s + (0.177 − 0.306i)25-s + (2.77 − 4.81i)29-s + 2.50i·31-s + (0.887 − 0.512i)35-s + 3.93i·37-s + (2.99 + 5.19i)41-s + (−1.54 − 2.67i)43-s + (7.99 + 4.61i)47-s + ⋯
L(s)  = 1  + (0.896 − 0.517i)5-s + 0.167·7-s + 1.31i·11-s + (0.527 + 0.304i)13-s + (0.428 − 0.247i)17-s + (0.634 − 0.772i)19-s + (−0.0202 − 0.0116i)23-s + (0.0354 − 0.0613i)25-s + (0.515 − 0.893i)29-s + 0.450i·31-s + (0.150 − 0.0866i)35-s + 0.647i·37-s + (0.468 + 0.810i)41-s + (−0.235 − 0.407i)43-s + (1.16 + 0.673i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.985 - 0.169i$
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: \((2,\ 2736,\ (\ :1/2),\ 0.985 - 0.169i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.380107394\)
\(L(\frac12)\) \(\approx\) \(2.380107394\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.76 + 3.36i)T \)
good5 \( 1 + (-2.00 + 1.15i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 - 0.442T + 7T^{2} \)
11 \( 1 - 4.35iT - 11T^{2} \)
13 \( 1 + (-1.90 - 1.09i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.76 + 1.02i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (0.0969 + 0.0559i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.77 + 4.81i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.50iT - 31T^{2} \)
37 \( 1 - 3.93iT - 37T^{2} \)
41 \( 1 + (-2.99 - 5.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.54 + 2.67i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-7.99 - 4.61i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.77 - 6.53i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.993 - 1.72i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.98 + 5.16i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.09 - 1.78i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.86 - 6.69i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (4.78 + 8.28i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.17 + 2.98i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 2.81iT - 83T^{2} \)
89 \( 1 + (5.87 - 10.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-10.7 + 6.20i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.015289993499941032551202672762, −8.099507056931492285736320675526, −7.31634957860681291759844475824, −6.53970063344357562043010713278, −5.71662166457364715735731630063, −4.91383991845969436593042957766, −4.32428989352376578896125704631, −3.02309511721681051219336296819, −2.01065874367336000414782692289, −1.12167826097870157752291231024, 0.933966506999798419273662369624, 2.08601173053292044635137622836, 3.15806372557436994535828913285, 3.78582276274618120098575247038, 5.16005295709215263320728404504, 5.84489053961541217549607739759, 6.28093633974158302024444277564, 7.28549588007400579214377639310, 8.149252873706636077414660039804, 8.736342286703098493189700307078

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