Properties

Label 2-10051-1.1-c1-0-3
Degree $2$
Conductor $10051$
Sign $1$
Analytic cond. $80.2576$
Root an. cond. $8.95866$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

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

\[\begin{aligned}\Lambda(s)=\mathstrut & 10051 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10051 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10051\)    =    \(19 \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(80.2576\)
Root analytic conductor: \(8.95866\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{10051} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 10051,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 + T \)
23 \( 1 \)
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 \( 1 - 3 T + p T^{2} \)
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 \( 1 - T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
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

−17.15469800593192, −16.79908099321222, −16.08249838015658, −15.60422665754080, −14.93590494702388, −14.35088165615151, −13.89111466404330, −12.72883175972338, −12.62919432869185, −12.01739368991090, −11.42193199764378, −10.93710697696941, −10.15301544798868, −9.816675087789643, −8.727202144477165, −8.216363967283194, −7.700281057438310, −7.103236981523145, −6.168443600863349, −5.337706785761346, −4.893765567572146, −4.499362749158335, −3.552618191500270, −2.814324951559212, −1.295814619755455, 0, 0, 1.295814619755455, 2.814324951559212, 3.552618191500270, 4.499362749158335, 4.893765567572146, 5.337706785761346, 6.168443600863349, 7.103236981523145, 7.700281057438310, 8.216363967283194, 8.727202144477165, 9.816675087789643, 10.15301544798868, 10.93710697696941, 11.42193199764378, 12.01739368991090, 12.62919432869185, 12.72883175972338, 13.89111466404330, 14.35088165615151, 14.93590494702388, 15.60422665754080, 16.08249838015658, 16.79908099321222, 17.15469800593192

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