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Properties

Label 2-2736-76.75-c1-0-45
Degree $2$
Conductor $2736$
Sign $-0.114 + 0.993i$
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  + 3.37·5-s − 0.792i·7-s + 0.792i·11-s − 5.04i·13-s − 5.37·17-s + (−4 + 1.73i)19-s − 8.51i·23-s + 6.37·25-s − 10.0i·29-s − 8.74·31-s − 2.67i·35-s − 5.04i·37-s − 6.92i·41-s + 9.30i·43-s − 4.25i·47-s + ⋯
L(s)  = 1  + 1.50·5-s − 0.299i·7-s + 0.238i·11-s − 1.40i·13-s − 1.30·17-s + (−0.917 + 0.397i)19-s − 1.77i·23-s + 1.27·25-s − 1.87i·29-s − 1.57·31-s − 0.451i·35-s − 0.829i·37-s − 1.08i·41-s + 1.41i·43-s − 0.620i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.114 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.114 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.114 + 0.993i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (2431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ -0.114 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.804651397\)
\(L(\frac12)\) \(\approx\) \(1.804651397\)
\(L(\frac{3}{2})\) 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 \( 1 + (4 - 1.73i)T \)
good5 \( 1 - 3.37T + 5T^{2} \)
7 \( 1 + 0.792iT - 7T^{2} \)
11 \( 1 - 0.792iT - 11T^{2} \)
13 \( 1 + 5.04iT - 13T^{2} \)
17 \( 1 + 5.37T + 17T^{2} \)
23 \( 1 + 8.51iT - 23T^{2} \)
29 \( 1 + 10.0iT - 29T^{2} \)
31 \( 1 + 8.74T + 31T^{2} \)
37 \( 1 + 5.04iT - 37T^{2} \)
41 \( 1 + 6.92iT - 41T^{2} \)
43 \( 1 - 9.30iT - 43T^{2} \)
47 \( 1 + 4.25iT - 47T^{2} \)
53 \( 1 - 3.16iT - 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 - 5.37T + 61T^{2} \)
67 \( 1 - 9.48T + 67T^{2} \)
71 \( 1 + 4T + 71T^{2} \)
73 \( 1 + 4.11T + 73T^{2} \)
79 \( 1 + 4.74T + 79T^{2} \)
83 \( 1 + 3.46iT - 83T^{2} \)
89 \( 1 - 13.2iT - 89T^{2} \)
97 \( 1 - 13.2iT - 97T^{2} \)
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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

−8.738259650789770417521455968783, −7.931368400225602991188221365644, −6.96917359861835909732783534891, −6.17333231775383399717901451656, −5.72697938976650771416221653942, −4.75018643528261297487889780066, −3.90094299011055597615512525627, −2.48092740852513496844548347704, −2.08839203693154512691743822417, −0.52120653099828986500041316356, 1.66459791690092712678203934137, 2.07572990849311684487837238816, 3.28570031160636039478597904318, 4.42166336276566601083519562996, 5.26651074123904024079015086202, 5.95289627906522139328631542909, 6.72697693678296635596078840793, 7.22623091861027558210915984260, 8.750584232157027384351460369670, 8.948759579642966692803447230130

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