Properties

Label 2-381150-1.1-c1-0-400
Degree $2$
Conductor $381150$
Sign $-1$
Analytic cond. $3043.49$
Root an. cond. $55.1679$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 7-s + 8-s − 13-s + 14-s + 16-s − 3·17-s + 4·19-s − 3·23-s − 26-s + 28-s + 3·29-s + 5·31-s + 32-s − 3·34-s + 10·37-s + 4·38-s + 9·41-s − 43-s − 3·46-s + 49-s − 52-s + 9·53-s + 56-s + 3·58-s − 9·59-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s − 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.727·17-s + 0.917·19-s − 0.625·23-s − 0.196·26-s + 0.188·28-s + 0.557·29-s + 0.898·31-s + 0.176·32-s − 0.514·34-s + 1.64·37-s + 0.648·38-s + 1.40·41-s − 0.152·43-s − 0.442·46-s + 1/7·49-s − 0.138·52-s + 1.23·53-s + 0.133·56-s + 0.393·58-s − 1.17·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 381150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 381150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(381150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(3043.49\)
Root analytic conductor: \(55.1679\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{381150} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 381150,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 - T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 - T \)
11 \( 1 \)
good13 \( 1 + T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 + 11 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−12.60113083422119, −12.14487842238803, −12.02336929180955, −11.34617665529785, −10.98766482538537, −10.63765435082784, −10.00621949902745, −9.478707124562973, −9.276464628526203, −8.451144334191333, −8.058887433008217, −7.653106602900445, −7.205774711798738, −6.485078081406124, −6.339182649662990, −5.584280366526552, −5.309534182790650, −4.613872313990015, −4.251825819897504, −3.916971867489447, −3.020396553915879, −2.666334283655824, −2.238463741167533, −1.379403539985903, −0.9557618517437279, 0, 0.9557618517437279, 1.379403539985903, 2.238463741167533, 2.666334283655824, 3.020396553915879, 3.916971867489447, 4.251825819897504, 4.613872313990015, 5.309534182790650, 5.584280366526552, 6.339182649662990, 6.485078081406124, 7.205774711798738, 7.653106602900445, 8.058887433008217, 8.451144334191333, 9.276464628526203, 9.478707124562973, 10.00621949902745, 10.63765435082784, 10.98766482538537, 11.34617665529785, 12.02336929180955, 12.14487842238803, 12.60113083422119

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