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Properties

Label 2-2736-57.50-c1-0-20
Degree $2$
Conductor $2736$
Sign $0.600 - 0.799i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.600 - 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.600 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.600 - 0.799i$
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: \((2,\ 2736,\ (\ :1/2),\ 0.600 - 0.799i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.245150866\)
\(L(\frac12)\) \(\approx\) \(2.245150866\)
\(L(\frac{3}{2})\) 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

−8.951703111867673503078702904492, −8.141746259298938395706996369747, −7.36262141812155502385123855856, −6.92722819172550318490690710925, −5.55030930933979473501242946355, −5.00745894211047923975235024604, −4.53292551631563354012396507438, −3.20165926242682729976985538956, −1.96230387895430539692664771612, −1.42424820199624414917069757087, 0.74620476490508314624978228702, 1.99781584331197555497295916155, 2.84718886001384022433994239774, 3.98931447689414135389275425362, 4.81226828671111579549847094280, 5.85091007390287389818891282118, 6.10763195428405817302696192878, 7.33400383599136827091903395882, 8.032936465600252288268252405964, 8.639347191758585263856696092508

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