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Properties

Label 2-2736-57.8-c1-0-6
Degree $2$
Conductor $2736$
Sign $-0.168 - 0.985i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.168 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.168 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.168 - 0.985i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (1889, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ -0.168 - 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8774365674\)
\(L(\frac12)\) \(\approx\) \(0.8774365674\)
\(L(\frac{3}{2})\) 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

−9.032400207374728804333839702772, −7.989740916166054266491737722090, −7.81303942797420615541813588882, −6.86210087375721640960144106918, −5.94784019046549578438784127505, −4.97549888036399239900606138550, −4.31212357665266477252100094217, −3.59827315259314822621847168130, −2.34572267883555563727396381105, −1.18548958650470882389318546987, 0.31129780536056276204859173882, 1.75003799099346744756082748895, 3.30750145588817691398860626131, 3.43912656360992014512930269741, 4.67305296993087538403491144801, 5.49781879860218512076235907111, 6.48741940710086715089922313466, 7.05095720707455746300594516689, 7.922207828043312399224430206898, 8.587751182016544572003946151138

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