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Properties

Label 1-4725-4725.1921-r1-0-0
Degree $1$
Conductor $4725$
Sign $0.999 + 0.0173i$
Analytic cond. $507.771$
Root an. cond. $507.771$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.961 + 0.275i)2-s + (0.848 + 0.529i)4-s + (0.669 + 0.743i)8-s + (−0.241 − 0.970i)11-s + (0.241 − 0.970i)13-s + (0.438 + 0.898i)16-s + (−0.669 − 0.743i)17-s + (0.978 − 0.207i)19-s + (0.0348 − 0.999i)22-s + (0.559 + 0.829i)23-s + (0.5 − 0.866i)26-s + (−0.374 + 0.927i)29-s + (0.374 + 0.927i)31-s + (0.173 + 0.984i)32-s + (−0.438 − 0.898i)34-s + ⋯
L(s)  = 1  + (0.961 + 0.275i)2-s + (0.848 + 0.529i)4-s + (0.669 + 0.743i)8-s + (−0.241 − 0.970i)11-s + (0.241 − 0.970i)13-s + (0.438 + 0.898i)16-s + (−0.669 − 0.743i)17-s + (0.978 − 0.207i)19-s + (0.0348 − 0.999i)22-s + (0.559 + 0.829i)23-s + (0.5 − 0.866i)26-s + (−0.374 + 0.927i)29-s + (0.374 + 0.927i)31-s + (0.173 + 0.984i)32-s + (−0.438 − 0.898i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4725 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4725 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4725\)    =    \(3^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.999 + 0.0173i$
Analytic conductor: \(507.771\)
Root analytic conductor: \(507.771\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4725} (1921, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4725,\ (1:\ ),\ 0.999 + 0.0173i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.059938807 + 0.04396295704i\)
\(L(\frac12)\) \(\approx\) \(5.059938807 + 0.04396295704i\)
\(L(1)\) \(\approx\) \(2.010298210 + 0.2147873058i\)
\(L(1)\) \(\approx\) \(2.010298210 + 0.2147873058i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.961 + 0.275i)T \)
11 \( 1 + (-0.241 - 0.970i)T \)
13 \( 1 + (0.241 - 0.970i)T \)
17 \( 1 + (-0.669 - 0.743i)T \)
19 \( 1 + (0.978 - 0.207i)T \)
23 \( 1 + (0.559 + 0.829i)T \)
29 \( 1 + (-0.374 + 0.927i)T \)
31 \( 1 + (0.374 + 0.927i)T \)
37 \( 1 + (-0.809 + 0.587i)T \)
41 \( 1 + (0.241 - 0.970i)T \)
43 \( 1 + (-0.939 + 0.342i)T \)
47 \( 1 + (0.374 - 0.927i)T \)
53 \( 1 + (0.669 - 0.743i)T \)
59 \( 1 + (0.719 + 0.694i)T \)
61 \( 1 + (0.241 + 0.970i)T \)
67 \( 1 + (-0.615 + 0.788i)T \)
71 \( 1 + (0.669 - 0.743i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + (-0.615 - 0.788i)T \)
83 \( 1 + (-0.848 + 0.529i)T \)
89 \( 1 + (0.104 - 0.994i)T \)
97 \( 1 + (0.615 + 0.788i)T \)
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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

−18.114602269914891119640418396716, −17.13450519853733063359955354571, −16.618349830458184905079894582407, −15.67410250436120862039899820935, −15.32398187167668649354059237001, −14.54920708522438142005650420417, −13.95517788114173495339917065296, −13.238230061429131786190633853725, −12.67850551386814482492774983053, −11.97987765957086939715600303603, −11.35100799164115342994605636158, −10.72768212996754079237009496882, −9.878603646949736322939255353986, −9.37626865348054649720800353321, −8.3036303085656910868027506559, −7.45412037555988900805564331267, −6.780741001761375730668587038064, −6.18018597956040943725756499565, −5.348954937572721824350920048539, −4.51543251354551218729509166457, −4.12230433869682487942869208380, −3.2052162396068108580356000417, −2.24177193119745646402646975587, −1.800276318042384504786292658203, −0.725605711667608702955714410461, 0.60386937129498683573452404475, 1.53263374273575709502749927102, 2.687685270050433688045349095723, 3.222670868505889643905284580434, 3.78900414000884093827534101999, 5.07659732389798290659022531618, 5.24584402354749126552782999245, 6.034076709571829893854182726096, 7.05159172481539197781882165104, 7.32514583248659886484335514786, 8.440109019348652232812025640914, 8.80555949333161268465225077600, 10.03687174508037203436755497997, 10.71605501369058585339965069611, 11.416143235045154289626267774963, 11.8999486125537938541055787326, 12.800747791737802326640009106927, 13.53180394931867349638054294488, 13.69933162663554275917538294045, 14.64125690590058128619657829012, 15.36508459349671847196560357663, 15.90586548511494040430718733031, 16.36207500187627232220450933924, 17.25279349647503458043081505784, 17.90359087482442812070254688441

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