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Properties

Label 2-2736-228.11-c1-0-34
Degree $2$
Conductor $2736$
Sign $-0.238 + 0.971i$
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  + (2.03 + 1.17i)5-s − 4.52i·7-s − 2.20·11-s + (−1.93 − 3.35i)13-s + (5.29 + 3.05i)17-s + (−2.75 − 3.37i)19-s + (0.557 + 0.966i)23-s + (0.270 + 0.468i)25-s + (3.51 − 2.02i)29-s + 6.92i·31-s + (5.33 − 9.23i)35-s + 10.9·37-s + (−8.80 − 5.08i)41-s + (−9.76 − 5.63i)43-s + (−6.43 − 11.1i)47-s + ⋯
L(s)  = 1  + (0.911 + 0.526i)5-s − 1.71i·7-s − 0.666·11-s + (−0.536 − 0.929i)13-s + (1.28 + 0.740i)17-s + (−0.632 − 0.774i)19-s + (0.116 + 0.201i)23-s + (0.0541 + 0.0937i)25-s + (0.652 − 0.376i)29-s + 1.24i·31-s + (0.900 − 1.56i)35-s + 1.80·37-s + (−1.37 − 0.794i)41-s + (−1.48 − 0.859i)43-s + (−0.938 − 1.62i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.238 + 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.238 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.238 + 0.971i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ -0.238 + 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.618470842\)
\(L(\frac12)\) \(\approx\) \(1.618470842\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.75 + 3.37i)T \)
good5 \( 1 + (-2.03 - 1.17i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + 4.52iT - 7T^{2} \)
11 \( 1 + 2.20T + 11T^{2} \)
13 \( 1 + (1.93 + 3.35i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-5.29 - 3.05i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-0.557 - 0.966i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.51 + 2.02i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.92iT - 31T^{2} \)
37 \( 1 - 10.9T + 37T^{2} \)
41 \( 1 + (8.80 + 5.08i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (9.76 + 5.63i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.43 + 11.1i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (7.32 - 4.23i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.66 + 6.35i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.719 + 1.24i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.28 + 3.05i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.87 - 6.70i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.705 + 1.22i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.92 + 1.11i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 11.4T + 83T^{2} \)
89 \( 1 + (2.44 - 1.40i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.76 - 6.52i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.368701915720879895035802331398, −7.86753825270445201281542331347, −6.98727866208757171958372013933, −6.49756540384855044712793902126, −5.44889605451784094704365077986, −4.79184932128372491367901355344, −3.64992372726148757777809901378, −2.93528705621843611918457083246, −1.75507802513681958350585249277, −0.49877301197056449558465303666, 1.51136635924389320865231580111, 2.36361681592590606684704939881, 3.10616354159616945071857776126, 4.67764150959190644837623181380, 5.16634940694565903191721335133, 5.97155679335808727996854005361, 6.43681324295028092824130968798, 7.78825889567053911231438846810, 8.288701288219060985365154547573, 9.211554055760597391416543026725

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