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Properties

Label 1-2555-2555.2554-r1-0-0
Degree $1$
Conductor $2555$
Sign $1$
Analytic cond. $274.572$
Root an. cond. $274.572$
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  − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s − 11-s + 12-s − 13-s + 16-s − 17-s − 18-s − 19-s + 22-s − 23-s − 24-s + 26-s + 27-s − 29-s + 31-s − 32-s − 33-s + 34-s + 36-s − 37-s + 38-s − 39-s + ⋯
L(s)  = 1  − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s − 11-s + 12-s − 13-s + 16-s − 17-s − 18-s − 19-s + 22-s − 23-s − 24-s + 26-s + 27-s − 29-s + 31-s − 32-s − 33-s + 34-s + 36-s − 37-s + 38-s − 39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2555 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2555 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2555\)    =    \(5 \cdot 7 \cdot 73\)
Sign: $1$
Analytic conductor: \(274.572\)
Root analytic conductor: \(274.572\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2555} (2554, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 2555,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9835961901\)
\(L(\frac12)\) \(\approx\) \(0.9835961901\)
\(L(1)\) \(\approx\) \(0.7458228185\)
\(L(1)\) \(\approx\) \(0.7458228185\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
73 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + T \)
11 \( 1 - T \)
13 \( 1 - T \)
17 \( 1 - T \)
19 \( 1 - T \)
23 \( 1 - T \)
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 \( 1 + T \)
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

−19.42100603431393838739942205925, −18.53146613382033796490160885906, −17.98175520588604063098561195514, −17.2050608424332366451036576862, −16.41216370070195634819251184772, −15.50214533792589653040960224670, −15.268643391005377330161511244975, −14.41039625823134060666353116148, −13.50470792837175033356474852698, −12.75383466902032202094314684959, −12.05331783298862045408294501841, −11.03335300006393066399617291514, −10.23337099220260971439850406781, −9.84960430000007939699476030465, −8.8794613943820341311513131751, −8.38412687906959218364066426629, −7.64904428295506790558138278165, −7.045398354214119910590042560881, −6.20404638048430544768014516503, −5.05115564229776642677293604455, −4.09564952256859738843351837637, −3.0555939617817157756560221175, −2.25301033498651829145129958426, −1.86864713673285569464797785637, −0.39213401708952462606545528564, 0.39213401708952462606545528564, 1.86864713673285569464797785637, 2.25301033498651829145129958426, 3.0555939617817157756560221175, 4.09564952256859738843351837637, 5.05115564229776642677293604455, 6.20404638048430544768014516503, 7.045398354214119910590042560881, 7.64904428295506790558138278165, 8.38412687906959218364066426629, 8.8794613943820341311513131751, 9.84960430000007939699476030465, 10.23337099220260971439850406781, 11.03335300006393066399617291514, 12.05331783298862045408294501841, 12.75383466902032202094314684959, 13.50470792837175033356474852698, 14.41039625823134060666353116148, 15.268643391005377330161511244975, 15.50214533792589653040960224670, 16.41216370070195634819251184772, 17.2050608424332366451036576862, 17.98175520588604063098561195514, 18.53146613382033796490160885906, 19.42100603431393838739942205925

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