Properties

Label 2-2736-19.11-c1-0-12
Degree $2$
Conductor $2736$
Sign $-0.0977 - 0.995i$
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  + 7-s + (3.5 + 6.06i)13-s + (−4 + 1.73i)19-s + (2.5 + 4.33i)25-s − 11·31-s − 37-s + (2.5 − 4.33i)43-s − 6·49-s + (6.5 + 11.2i)61-s + (2.5 + 4.33i)67-s + (3.5 − 6.06i)73-s + (−6.5 + 11.2i)79-s + (3.5 + 6.06i)91-s + (−7 + 12.1i)97-s + 13·103-s + ⋯
L(s)  = 1  + 0.377·7-s + (0.970 + 1.68i)13-s + (−0.917 + 0.397i)19-s + (0.5 + 0.866i)25-s − 1.97·31-s − 0.164·37-s + (0.381 − 0.660i)43-s − 0.857·49-s + (0.832 + 1.44i)61-s + (0.305 + 0.529i)67-s + (0.409 − 0.709i)73-s + (−0.731 + 1.26i)79-s + (0.366 + 0.635i)91-s + (−0.710 + 1.23i)97-s + 1.28·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0977 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0977 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.0977 - 0.995i$
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: \((2,\ 2736,\ (\ :1/2),\ -0.0977 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.486755136\)
\(L(\frac12)\) \(\approx\) \(1.486755136\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 - 4.33i)T^{2} \)
7 \( 1 - T + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + (-3.5 - 6.06i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 11T + 31T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.5 + 4.33i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.5 - 11.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.5 - 4.33i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3.5 + 6.06i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.5 - 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7 - 12.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.909651498756595813186719366062, −8.481380592290244429273515498935, −7.40441181731652690071841014593, −6.79812311692226590575953247065, −5.99249343042853713846521599854, −5.14904996723655646782918689430, −4.15856331671155656265568194913, −3.61025810929865884953444879209, −2.18232044308614782165028871657, −1.40860371051596851626892960750, 0.48344629813890547028717724100, 1.78167645699854059480313143426, 2.93321309807980855637419691174, 3.75754052939657342197403457249, 4.75148266802816772793363610123, 5.55845607058151059496893826361, 6.25126405539364527328459067714, 7.14050091451346010672859184291, 8.067176148519715323875600612147, 8.443902891164979404425085391891

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