Properties

Label 2-2736-19.18-c2-0-20
Degree $2$
Conductor $2736$
Sign $-0.684 - 0.729i$
Analytic cond. $74.5506$
Root an. cond. $8.63426$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·7-s + 13.8i·13-s + (13 + 13.8i)19-s − 25·25-s − 41.5i·31-s + 69.2i·37-s + 22·43-s − 45·49-s − 74·61-s − 55.4i·67-s − 46·73-s − 69.2i·79-s + 27.7i·91-s + 193. i·97-s + 69.2i·103-s + ⋯
L(s)  = 1  + 0.285·7-s + 1.06i·13-s + (0.684 + 0.729i)19-s − 25-s − 1.34i·31-s + 1.87i·37-s + 0.511·43-s − 0.918·49-s − 1.21·61-s − 0.827i·67-s − 0.630·73-s − 0.876i·79-s + 0.304i·91-s + 1.99i·97-s + 0.672i·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.684 - 0.729i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.684 - 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.684 - 0.729i$
Analytic conductor: \(74.5506\)
Root analytic conductor: \(8.63426\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (721, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1),\ -0.684 - 0.729i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.091433785\)
\(L(\frac12)\) \(\approx\) \(1.091433785\)
\(L(2)\) 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 \( 1 + (-13 - 13.8i)T \)
good5 \( 1 + 25T^{2} \)
7 \( 1 - 2T + 49T^{2} \)
11 \( 1 + 121T^{2} \)
13 \( 1 - 13.8iT - 169T^{2} \)
17 \( 1 + 289T^{2} \)
23 \( 1 + 529T^{2} \)
29 \( 1 - 841T^{2} \)
31 \( 1 + 41.5iT - 961T^{2} \)
37 \( 1 - 69.2iT - 1.36e3T^{2} \)
41 \( 1 - 1.68e3T^{2} \)
43 \( 1 - 22T + 1.84e3T^{2} \)
47 \( 1 + 2.20e3T^{2} \)
53 \( 1 - 2.80e3T^{2} \)
59 \( 1 - 3.48e3T^{2} \)
61 \( 1 + 74T + 3.72e3T^{2} \)
67 \( 1 + 55.4iT - 4.48e3T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 + 46T + 5.32e3T^{2} \)
79 \( 1 + 69.2iT - 6.24e3T^{2} \)
83 \( 1 + 6.88e3T^{2} \)
89 \( 1 - 7.92e3T^{2} \)
97 \( 1 - 193. iT - 9.40e3T^{2} \)
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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

−9.013144266978583072329578246929, −8.007361810268010574227573822431, −7.60449867459815183143251332094, −6.54433168328556523433031073445, −5.96461097547096894978730221148, −4.96643410292395633095126028443, −4.21772121507232232250917605625, −3.34847740832089068041818314032, −2.18229001061301318026477935462, −1.28991481228132251366326940313, 0.25321990779208096679995880352, 1.44187198508490336775537557177, 2.62762603159696652539935684562, 3.45710881541466887365885444231, 4.45582537467749790195219789099, 5.34621053624017788457919020461, 5.89989611881296685569338479759, 7.00529757397510823557941474697, 7.59675905822747893054517567598, 8.334250963897739631860892092402

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