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Properties

Label 2-2736-19.11-c1-0-45
Degree $2$
Conductor $2736$
Sign $-0.425 + 0.905i$
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  + (1.82 − 3.15i)5-s + 1.64·7-s + 0.645·11-s + (−1 − 1.73i)13-s + (−4.32 − 0.559i)19-s + (−1.82 − 3.15i)23-s + (−4.14 − 7.18i)25-s + (−1.82 − 3.15i)29-s + 0.354·31-s + (3 − 5.19i)35-s + 5.64·37-s + (5.14 − 8.91i)41-s + (0.354 − 0.613i)43-s + (−4.82 − 8.35i)47-s − 4.29·49-s + ⋯
L(s)  = 1  + (0.815 − 1.41i)5-s + 0.622·7-s + 0.194·11-s + (−0.277 − 0.480i)13-s + (−0.991 − 0.128i)19-s + (−0.380 − 0.658i)23-s + (−0.829 − 1.43i)25-s + (−0.338 − 0.586i)29-s + 0.0636·31-s + (0.507 − 0.878i)35-s + 0.928·37-s + (0.803 − 1.39i)41-s + (0.0540 − 0.0935i)43-s + (−0.703 − 1.21i)47-s − 0.613·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.425 + 0.905i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.425 + 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.425 + 0.905i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (1873, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ -0.425 + 0.905i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.924519236\)
\(L(\frac12)\) \(\approx\) \(1.924519236\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.32 + 0.559i)T \)
good5 \( 1 + (-1.82 + 3.15i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 - 1.64T + 7T^{2} \)
11 \( 1 - 0.645T + 11T^{2} \)
13 \( 1 + (1 + 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (1.82 + 3.15i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.82 + 3.15i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 0.354T + 31T^{2} \)
37 \( 1 - 5.64T + 37T^{2} \)
41 \( 1 + (-5.14 + 8.91i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.354 + 0.613i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.82 + 8.35i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.29 - 7.43i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.96 - 6.87i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.46 - 12.9i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.32 + 4.02i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.64 + 11.5i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (6.14 - 10.6i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2 - 3.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 7.93T + 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.14 + 12.3i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.608429892863232288030896976826, −8.074704963986473410853393969332, −7.11166484669905806708141769172, −5.99644108475227599309039960351, −5.54221969841503472916907689276, −4.62001640863046373575699539564, −4.12103743839861962788306122632, −2.52759750500203615321565477079, −1.71474862467648282091139088676, −0.59453285689327340530651658940, 1.61448955139252002146015092117, 2.37493411796785675003607466866, 3.30579099035312938934230556953, 4.31840373019551003757016778795, 5.26256474916239944741246221604, 6.27625408463510717474122726260, 6.57095107165577648874565977016, 7.54430003150297000591256217059, 8.164681743189404516145762289678, 9.272967267100967419703881175991

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