Properties

Label 1-1012-1012.1011-r1-0-0
Degree $1$
Conductor $1012$
Sign $1$
Analytic cond. $108.754$
Root an. cond. $108.754$
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  − 3-s − 5-s − 7-s + 9-s − 13-s + 15-s + 17-s − 19-s + 21-s + 25-s − 27-s − 29-s − 31-s + 35-s − 37-s + 39-s − 41-s − 43-s − 45-s − 47-s + 49-s − 51-s − 53-s + 57-s − 59-s + 61-s − 63-s + ⋯
L(s)  = 1  − 3-s − 5-s − 7-s + 9-s − 13-s + 15-s + 17-s − 19-s + 21-s + 25-s − 27-s − 29-s − 31-s + 35-s − 37-s + 39-s − 41-s − 43-s − 45-s − 47-s + 49-s − 51-s − 53-s + 57-s − 59-s + 61-s − 63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1012\)    =    \(2^{2} \cdot 11 \cdot 23\)
Sign: $1$
Analytic conductor: \(108.754\)
Root analytic conductor: \(108.754\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1012} (1011, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 1012,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04870024580\)
\(L(\frac12)\) \(\approx\) \(0.04870024580\)
\(L(1)\) \(\approx\) \(0.3950204758\)
\(L(1)\) \(\approx\) \(0.3950204758\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
23 \( 1 \)
good3 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 - T \)
13 \( 1 - T \)
17 \( 1 + T \)
19 \( 1 - T \)
29 \( 1 - T \)
31 \( 1 - T \)
37 \( 1 - T \)
41 \( 1 - T \)
43 \( 1 - T \)
47 \( 1 - T \)
53 \( 1 - T \)
59 \( 1 - T \)
61 \( 1 + T \)
67 \( 1 + T \)
71 \( 1 - T \)
73 \( 1 - T \)
79 \( 1 - T \)
83 \( 1 - T \)
89 \( 1 - T \)
97 \( 1 - T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.736114788312010332020366442357, −20.622925934335468248105636703416, −19.7055223823745399107469714046, −18.97138417564053962930515190614, −18.555246096564749490587386570375, −17.24119391098984576249949940021, −16.719697087666148023882231624847, −16.0906429187151965972423479002, −15.260432942255408030334558656580, −14.56068176517078421782452581459, −13.109702240349699896297774945023, −12.54450584739198238283459817912, −11.94513279184025215970281069887, −11.10193587325115487102688849534, −10.230672534201348004051587683609, −9.56360743843928311920397302480, −8.35746741597925856612341031233, −7.28463458774917937101622600666, −6.826335033312680050928120810603, −5.747402044508024193530845353282, −4.90461665078782728657915435459, −3.928745557801679677069526183274, −3.13145005148115808349175463316, −1.63007179383371076139374007318, −0.10666881536242395641005015805, 0.10666881536242395641005015805, 1.63007179383371076139374007318, 3.13145005148115808349175463316, 3.928745557801679677069526183274, 4.90461665078782728657915435459, 5.747402044508024193530845353282, 6.826335033312680050928120810603, 7.28463458774917937101622600666, 8.35746741597925856612341031233, 9.56360743843928311920397302480, 10.230672534201348004051587683609, 11.10193587325115487102688849534, 11.94513279184025215970281069887, 12.54450584739198238283459817912, 13.109702240349699896297774945023, 14.56068176517078421782452581459, 15.260432942255408030334558656580, 16.0906429187151965972423479002, 16.719697087666148023882231624847, 17.24119391098984576249949940021, 18.555246096564749490587386570375, 18.97138417564053962930515190614, 19.7055223823745399107469714046, 20.622925934335468248105636703416, 21.736114788312010332020366442357

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