Properties

Label 2-2736-57.50-c1-0-2
Degree $2$
Conductor $2736$
Sign $-0.953 - 0.300i$
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  + (−1.22 + 0.707i)5-s + 3.44·7-s − 6.29i·11-s + (−2.17 − 1.25i)13-s + (−4.22 + 2.43i)17-s + (−4 + 1.73i)19-s + (−4.89 − 2.82i)23-s + (−1.50 + 2.59i)25-s + (1.22 − 2.12i)29-s + 9.43i·31-s + (−4.22 + 2.43i)35-s + 5.97i·37-s + (−2.94 − 5.10i)43-s + (−4.22 − 2.43i)47-s + 4.89·49-s + ⋯
L(s)  = 1  + (−0.547 + 0.316i)5-s + 1.30·7-s − 1.89i·11-s + (−0.603 − 0.348i)13-s + (−1.02 + 0.591i)17-s + (−0.917 + 0.397i)19-s + (−1.02 − 0.589i)23-s + (−0.300 + 0.519i)25-s + (0.227 − 0.393i)29-s + 1.69i·31-s + (−0.714 + 0.412i)35-s + 0.982i·37-s + (−0.449 − 0.779i)43-s + (−0.616 − 0.355i)47-s + 0.699·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 - 0.300i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.953 - 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.953 - 0.300i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ -0.953 - 0.300i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1374696819\)
\(L(\frac12)\) \(\approx\) \(0.1374696819\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.22 - 0.707i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 - 3.44T + 7T^{2} \)
11 \( 1 + 6.29iT - 11T^{2} \)
13 \( 1 + (2.17 + 1.25i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (4.22 - 2.43i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (4.89 + 2.82i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.22 + 2.12i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 9.43iT - 31T^{2} \)
37 \( 1 - 5.97iT - 37T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.94 + 5.10i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.22 + 2.43i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.44 - 4.24i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.77 - 8.27i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.724 - 1.25i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.84 + 3.37i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3 - 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.94 - 10.3i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.17 - 4.71i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 8.97iT - 83T^{2} \)
89 \( 1 + (6.12 - 10.6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.65 - 0.953i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.681960583076072009408516198919, −8.460698812580524309749569003345, −7.900891119267882792609317526484, −6.88691219826401011007682497965, −6.10298626515778721430123935003, −5.28931250779234475083679541220, −4.39480181769962949658977673957, −3.62940711054893405007587423713, −2.60255352649336640045602417744, −1.46532100779203454900093783724, 0.04183068726093175586842652065, 1.87481873176743822911907625510, 2.26964039749189624848372912621, 4.05252640568391062345891931900, 4.57413233666992765343874673778, 4.98239696342815774093503302745, 6.28498501606813475465846017659, 7.16328072978569562253946771807, 7.75127890871336963244926640234, 8.307682349834754712078687808634

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