Properties

Label 2-38e2-1.1-c3-0-32
Degree $2$
Conductor $1444$
Sign $1$
Analytic cond. $85.1987$
Root an. cond. $9.23031$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.37·3-s + 11.8·5-s − 26.4·7-s + 1.86·9-s − 49.8·11-s + 49.0·13-s + 63.7·15-s + 17.2·17-s − 142.·21-s + 166.·23-s + 15.6·25-s − 135.·27-s + 109.·29-s + 273.·31-s − 267.·33-s − 314.·35-s − 167.·37-s + 263.·39-s − 15.1·41-s + 413.·43-s + 22.0·45-s − 161.·47-s + 358.·49-s + 92.5·51-s + 490.·53-s − 591.·55-s + 335.·59-s + ⋯
L(s)  = 1  + 1.03·3-s + 1.06·5-s − 1.43·7-s + 0.0689·9-s − 1.36·11-s + 1.04·13-s + 1.09·15-s + 0.245·17-s − 1.47·21-s + 1.51·23-s + 0.125·25-s − 0.962·27-s + 0.699·29-s + 1.58·31-s − 1.41·33-s − 1.51·35-s − 0.742·37-s + 1.08·39-s − 0.0577·41-s + 1.46·43-s + 0.0731·45-s − 0.501·47-s + 1.04·49-s + 0.254·51-s + 1.27·53-s − 1.44·55-s + 0.741·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1444\)    =    \(2^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(85.1987\)
Root analytic conductor: \(9.23031\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1444,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(3.114793620\)
\(L(\frac12)\) \(\approx\) \(3.114793620\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
good3 \( 1 - 5.37T + 27T^{2} \)
5 \( 1 - 11.8T + 125T^{2} \)
7 \( 1 + 26.4T + 343T^{2} \)
11 \( 1 + 49.8T + 1.33e3T^{2} \)
13 \( 1 - 49.0T + 2.19e3T^{2} \)
17 \( 1 - 17.2T + 4.91e3T^{2} \)
23 \( 1 - 166.T + 1.21e4T^{2} \)
29 \( 1 - 109.T + 2.43e4T^{2} \)
31 \( 1 - 273.T + 2.97e4T^{2} \)
37 \( 1 + 167.T + 5.06e4T^{2} \)
41 \( 1 + 15.1T + 6.89e4T^{2} \)
43 \( 1 - 413.T + 7.95e4T^{2} \)
47 \( 1 + 161.T + 1.03e5T^{2} \)
53 \( 1 - 490.T + 1.48e5T^{2} \)
59 \( 1 - 335.T + 2.05e5T^{2} \)
61 \( 1 - 725.T + 2.26e5T^{2} \)
67 \( 1 + 497.T + 3.00e5T^{2} \)
71 \( 1 - 798.T + 3.57e5T^{2} \)
73 \( 1 + 311.T + 3.89e5T^{2} \)
79 \( 1 - 665.T + 4.93e5T^{2} \)
83 \( 1 + 372.T + 5.71e5T^{2} \)
89 \( 1 - 673.T + 7.04e5T^{2} \)
97 \( 1 - 960.T + 9.12e5T^{2} \)
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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

−9.044677785179141828645965037369, −8.609456163258963838504126165994, −7.64863563215340404060239761439, −6.63801105897354697924467317540, −5.94835414791555536078988212849, −5.13128106272326251727369212943, −3.66625614898023446721353897152, −2.88889493157808403279340589222, −2.34834151893248409147297584307, −0.811800537377448312204552249538, 0.811800537377448312204552249538, 2.34834151893248409147297584307, 2.88889493157808403279340589222, 3.66625614898023446721353897152, 5.13128106272326251727369212943, 5.94835414791555536078988212849, 6.63801105897354697924467317540, 7.64863563215340404060239761439, 8.609456163258963838504126165994, 9.044677785179141828645965037369

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