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Properties

Label 2-3743-3743.3742-c0-0-5
Degree $2$
Conductor $3743$
Sign $1$
Analytic cond. $1.86800$
Root an. cond. $1.36674$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 2·7-s − 9-s + 16-s + 19-s − 2·23-s + 25-s + 2·28-s + 36-s − 2·43-s + 2·47-s + 3·49-s + 2·61-s + 2·63-s − 64-s − 76-s + 81-s + 2·83-s + 2·92-s − 100-s + 2·101-s − 2·112-s + ⋯
L(s)  = 1  − 4-s − 2·7-s − 9-s + 16-s + 19-s − 2·23-s + 25-s + 2·28-s + 36-s − 2·43-s + 2·47-s + 3·49-s + 2·61-s + 2·63-s − 64-s − 76-s + 81-s + 2·83-s + 2·92-s − 100-s + 2·101-s − 2·112-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3743 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3743 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3743\)    =    \(19 \cdot 197\)
Sign: $1$
Analytic conductor: \(1.86800\)
Root analytic conductor: \(1.36674\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3743} (3742, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3743,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5288012107\)
\(L(\frac12)\) \(\approx\) \(0.5288012107\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 - T \)
197 \( 1 - T \)
good2 \( 1 + T^{2} \)
3 \( 1 + T^{2} \)
5 \( ( 1 - T )( 1 + T ) \)
7 \( ( 1 + T )^{2} \)
11 \( ( 1 - T )( 1 + T ) \)
13 \( 1 + T^{2} \)
17 \( ( 1 - T )( 1 + T ) \)
23 \( ( 1 + T )^{2} \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( 1 + T^{2} \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( ( 1 + T )^{2} \)
47 \( ( 1 - T )^{2} \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 - T )^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + T^{2} \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( 1 + T^{2} \)
83 \( ( 1 - T )^{2} \)
89 \( 1 + T^{2} \)
97 \( ( 1 - T )( 1 + T ) \)
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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

−8.824961843670571051039549933777, −8.125733582233496089586523136849, −7.19020830215393293961000449912, −6.30294028529091215700314489134, −5.78004250773463171790913763185, −5.02297458928626960221700252374, −3.80107612909684019858055894620, −3.41392227317376315526980507047, −2.48491200247477105610177028689, −0.59638695915697831471377860335, 0.59638695915697831471377860335, 2.48491200247477105610177028689, 3.41392227317376315526980507047, 3.80107612909684019858055894620, 5.02297458928626960221700252374, 5.78004250773463171790913763185, 6.30294028529091215700314489134, 7.19020830215393293961000449912, 8.125733582233496089586523136849, 8.824961843670571051039549933777

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