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Properties

Label 2-2736-228.83-c1-0-7
Degree $2$
Conductor $2736$
Sign $-0.425 - 0.905i$
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  + (0.420 − 0.242i)5-s − 1.92i·7-s + 0.610·11-s + (−0.889 + 1.54i)13-s + (−3.89 + 2.25i)17-s + (−1.30 + 4.15i)19-s + (0.121 − 0.209i)23-s + (−2.38 + 4.12i)25-s + (−5.18 − 2.99i)29-s + 0.423i·31-s + (−0.467 − 0.809i)35-s − 1.98·37-s + (−5.42 + 3.13i)41-s + (3.24 − 1.87i)43-s + (−2.88 + 4.99i)47-s + ⋯
L(s)  = 1  + (0.187 − 0.108i)5-s − 0.728i·7-s + 0.184·11-s + (−0.246 + 0.427i)13-s + (−0.945 + 0.546i)17-s + (−0.298 + 0.954i)19-s + (0.0252 − 0.0437i)23-s + (−0.476 + 0.825i)25-s + (−0.962 − 0.555i)29-s + 0.0759i·31-s + (−0.0789 − 0.136i)35-s − 0.326·37-s + (−0.847 + 0.489i)41-s + (0.495 − 0.286i)43-s + (−0.420 + 0.728i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.425 - 0.905i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.425 - 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.425 - 0.905i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (2591, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ -0.425 - 0.905i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8213986970\)
\(L(\frac12)\) \(\approx\) \(0.8213986970\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.30 - 4.15i)T \)
good5 \( 1 + (-0.420 + 0.242i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + 1.92iT - 7T^{2} \)
11 \( 1 - 0.610T + 11T^{2} \)
13 \( 1 + (0.889 - 1.54i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.89 - 2.25i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (-0.121 + 0.209i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (5.18 + 2.99i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 0.423iT - 31T^{2} \)
37 \( 1 + 1.98T + 37T^{2} \)
41 \( 1 + (5.42 - 3.13i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.24 + 1.87i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.88 - 4.99i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.85 + 1.64i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.16 - 3.75i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.59 - 2.75i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.13 - 2.38i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-7.60 - 13.1i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.34 - 5.78i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-13.3 + 7.71i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 8.99T + 83T^{2} \)
89 \( 1 + (8.14 + 4.70i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.18 - 5.52i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.100448435116157297558426951065, −8.297272634083590171304627124870, −7.53568518069405465638212808839, −6.79845737209437991300511178909, −6.06699423299216434327884757976, −5.19112852368429403390893288764, −4.16556590657552354507635942036, −3.68942642333013127388791876488, −2.29581116018623798765034466169, −1.37972093436697573627364033322, 0.25409911176577484264675925209, 1.95637634383497754182725113334, 2.66920283617399145911914780019, 3.72865072080569044764170980533, 4.80061900965952104608077518600, 5.40940220246293301354440336025, 6.36850169011323858771063316888, 6.95124476781509233573862235486, 7.87574384206409121682676283439, 8.707898882031265296286677026120

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