Properties

Label 2-1224-136.67-c0-0-0
Degree $2$
Conductor $1224$
Sign $1$
Analytic cond. $0.610855$
Root an. cond. $0.781572$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 1.41·5-s − 1.41·7-s − 8-s + 1.41·10-s + 1.41·14-s + 16-s + 17-s − 1.41·20-s + 1.41·23-s + 1.00·25-s − 1.41·28-s + 1.41·29-s + 1.41·31-s − 32-s − 34-s + 2.00·35-s + 1.41·37-s + 1.41·40-s − 2·43-s − 1.41·46-s + 1.00·49-s − 1.00·50-s + 1.41·56-s − 1.41·58-s − 1.41·61-s + ⋯
L(s)  = 1  − 2-s + 4-s − 1.41·5-s − 1.41·7-s − 8-s + 1.41·10-s + 1.41·14-s + 16-s + 17-s − 1.41·20-s + 1.41·23-s + 1.00·25-s − 1.41·28-s + 1.41·29-s + 1.41·31-s − 32-s − 34-s + 2.00·35-s + 1.41·37-s + 1.41·40-s − 2·43-s − 1.41·46-s + 1.00·49-s − 1.00·50-s + 1.41·56-s − 1.41·58-s − 1.41·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1224\)    =    \(2^{3} \cdot 3^{2} \cdot 17\)
Sign: $1$
Analytic conductor: \(0.610855\)
Root analytic conductor: \(0.781572\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1224} (883, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1224,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4400220352\)
\(L(\frac12)\) \(\approx\) \(0.4400220352\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
17 \( 1 - T \)
good5 \( 1 + 1.41T + T^{2} \)
7 \( 1 + 1.41T + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - 1.41T + T^{2} \)
29 \( 1 - 1.41T + T^{2} \)
31 \( 1 - 1.41T + T^{2} \)
37 \( 1 - 1.41T + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + 2T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + 1.41T + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + 1.41T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - 1.41T + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - 2T + T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.921096026502799667245284448088, −9.089320330054712159635694496889, −8.262372192001780137580380421882, −7.61618971070998471370021541179, −6.78644340089740919807353642420, −6.16568914155503284830000562469, −4.71575497748483378375653844011, −3.36709884565606494745634365404, −2.93934477872671429160988624196, −0.835860832016285951941877200020, 0.835860832016285951941877200020, 2.93934477872671429160988624196, 3.36709884565606494745634365404, 4.71575497748483378375653844011, 6.16568914155503284830000562469, 6.78644340089740919807353642420, 7.61618971070998471370021541179, 8.262372192001780137580380421882, 9.089320330054712159635694496889, 9.921096026502799667245284448088

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