Properties

Label 2-41971-1.1-c1-0-0
Degree $2$
Conductor $41971$
Sign $1$
Analytic cond. $335.140$
Root an. cond. $18.3068$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 2·4-s − 3·5-s − 7-s + 9-s − 3·11-s + 4·12-s + 4·13-s + 6·15-s + 4·16-s − 3·17-s − 19-s + 6·20-s + 2·21-s + 4·25-s + 4·27-s + 2·28-s − 6·29-s + 4·31-s + 6·33-s + 3·35-s − 2·36-s + 2·37-s − 8·39-s + 6·41-s + 43-s + 6·44-s + ⋯
L(s)  = 1  − 1.15·3-s − 4-s − 1.34·5-s − 0.377·7-s + 1/3·9-s − 0.904·11-s + 1.15·12-s + 1.10·13-s + 1.54·15-s + 16-s − 0.727·17-s − 0.229·19-s + 1.34·20-s + 0.436·21-s + 4/5·25-s + 0.769·27-s + 0.377·28-s − 1.11·29-s + 0.718·31-s + 1.04·33-s + 0.507·35-s − 1/3·36-s + 0.328·37-s − 1.28·39-s + 0.937·41-s + 0.152·43-s + 0.904·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 41971 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41971 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(41971\)    =    \(19 \cdot 47^{2}\)
Sign: $1$
Analytic conductor: \(335.140\)
Root analytic conductor: \(18.3068\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 41971,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6019982966\)
\(L(\frac12)\) \(\approx\) \(0.6019982966\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 + T \)
47 \( 1 \)
good2 \( 1 + p T^{2} \)
3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−14.82062993720606, −14.17554838271630, −13.50924567493881, −13.03471210825680, −12.72711230000074, −12.07151758943378, −11.59054147429658, −11.08182005207214, −10.69725351046066, −10.20465328287405, −9.376933256368812, −8.872329961180960, −8.300762533956756, −7.840381302542135, −7.305031007172905, −6.411634217398548, −6.066246078508152, −5.369590393183792, −4.805546356798797, −4.314432567977842, −3.679847158564805, −3.208837694698318, −2.132348298031530, −0.7432226443089781, −0.4915439028913777, 0.4915439028913777, 0.7432226443089781, 2.132348298031530, 3.208837694698318, 3.679847158564805, 4.314432567977842, 4.805546356798797, 5.369590393183792, 6.066246078508152, 6.411634217398548, 7.305031007172905, 7.840381302542135, 8.300762533956756, 8.872329961180960, 9.376933256368812, 10.20465328287405, 10.69725351046066, 11.08182005207214, 11.59054147429658, 12.07151758943378, 12.72711230000074, 13.03471210825680, 13.50924567493881, 14.17554838271630, 14.82062993720606

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