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Properties

Label 1-1860-1860.1859-r1-0-0
Degree $1$
Conductor $1860$
Sign $1$
Analytic cond. $199.884$
Root an. cond. $199.884$
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  + 7-s − 11-s + 13-s − 17-s − 19-s + 23-s + 29-s + 37-s − 41-s − 43-s − 47-s + 49-s − 53-s + 59-s − 61-s + 67-s + 71-s + 73-s − 77-s + 79-s + 83-s + 89-s + 91-s − 97-s + ⋯
L(s)  = 1  + 7-s − 11-s + 13-s − 17-s − 19-s + 23-s + 29-s + 37-s − 41-s − 43-s − 47-s + 49-s − 53-s + 59-s − 61-s + 67-s + 71-s + 73-s − 77-s + 79-s + 83-s + 89-s + 91-s − 97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1860\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 31\)
Sign: $1$
Analytic conductor: \(199.884\)
Root analytic conductor: \(199.884\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1860} (1859, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 1860,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.300899290\)
\(L(\frac12)\) \(\approx\) \(2.300899290\)
\(L(1)\) \(\approx\) \(1.165502966\)
\(L(1)\) \(\approx\) \(1.165502966\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
31 \( 1 \)
good7 \( 1 + T \)
11 \( 1 - T \)
13 \( 1 + T \)
17 \( 1 - T \)
19 \( 1 - T \)
23 \( 1 + T \)
29 \( 1 + T \)
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

−20.06405066128818787301690164837, −19.12054067354123213934577429784, −18.32579623614432185540626721444, −17.878683133186906548961135853666, −17.08188651271173352946434891660, −16.24975652899715816353916161478, −15.34578565072898762894739359976, −14.99551835549859570269735407310, −13.92286214171722231008119405181, −13.29604472938635757135897977545, −12.649432769852087481419393239928, −11.52547176745411883277093687457, −10.96879009437601786936428853150, −10.434803713913328657557294010072, −9.30147303617279849882574710201, −8.2927062916976801118111929248, −8.16845195998352100864619510494, −6.889015776767079471839733364971, −6.23507484627342712087448685708, −5.07711388995961310538395184207, −4.65455713421514949836556877757, −3.5707249775972946769022419508, −2.51782927185210202064238951710, −1.71284001225451453111575785706, −0.62618185781168899946405160978, 0.62618185781168899946405160978, 1.71284001225451453111575785706, 2.51782927185210202064238951710, 3.5707249775972946769022419508, 4.65455713421514949836556877757, 5.07711388995961310538395184207, 6.23507484627342712087448685708, 6.889015776767079471839733364971, 8.16845195998352100864619510494, 8.2927062916976801118111929248, 9.30147303617279849882574710201, 10.434803713913328657557294010072, 10.96879009437601786936428853150, 11.52547176745411883277093687457, 12.649432769852087481419393239928, 13.29604472938635757135897977545, 13.92286214171722231008119405181, 14.99551835549859570269735407310, 15.34578565072898762894739359976, 16.24975652899715816353916161478, 17.08188651271173352946434891660, 17.878683133186906548961135853666, 18.32579623614432185540626721444, 19.12054067354123213934577429784, 20.06405066128818787301690164837

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