Properties

Label 2-736-184.45-c0-0-0
Degree $2$
Conductor $736$
Sign $-0.5 - 0.866i$
Analytic cond. $0.367311$
Root an. cond. $0.606062$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73i·3-s − 1.99·9-s + 1.73i·13-s + 23-s − 25-s − 1.73i·27-s − 1.73i·29-s − 31-s − 2.99·39-s + 41-s + 47-s + 49-s + 1.73i·69-s + 71-s + 73-s + ⋯
L(s)  = 1  + 1.73i·3-s − 1.99·9-s + 1.73i·13-s + 23-s − 25-s − 1.73i·27-s − 1.73i·29-s − 31-s − 2.99·39-s + 41-s + 47-s + 49-s + 1.73i·69-s + 71-s + 73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(736\)    =    \(2^{5} \cdot 23\)
Sign: $-0.5 - 0.866i$
Analytic conductor: \(0.367311\)
Root analytic conductor: \(0.606062\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{736} (689, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 736,\ (\ :0),\ -0.5 - 0.866i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8931733762\)
\(L(\frac12)\) \(\approx\) \(0.8931733762\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
23 \( 1 - T \)
good3 \( 1 - 1.73iT - T^{2} \)
5 \( 1 + T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - 1.73iT - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T^{2} \)
29 \( 1 + 1.73iT - T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - T + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T + T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T^{2} \)
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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

−10.89402246031875443791038589188, −9.835232259325409016958099415593, −9.350202908978857777754881799845, −8.708828689418450131342682296080, −7.47883361374290582131305253689, −6.26107439813339848174074069738, −5.31333819537829420019556322532, −4.29967861207433575666102926245, −3.81575594072983673035526441638, −2.37047274892109116800875543142, 1.01993668316202868155419309448, 2.37949505044292066581168201913, 3.42029591480502841622056557008, 5.27969799580445432496784259886, 5.92252044772329237695063413454, 7.02560275970692749895830921511, 7.57143381730088669121373194850, 8.337373881550224818091890852775, 9.214522955146016150464561862211, 10.55031157396877941342041929617

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