Properties

Label 1-42e2-1764.463-r1-0-0
Degree $1$
Conductor $1764$
Sign $0.915 - 0.401i$
Analytic cond. $189.568$
Root an. cond. $189.568$
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  + (0.0747 + 0.997i)5-s + (−0.365 − 0.930i)11-s + (−0.988 − 0.149i)13-s + (−0.222 + 0.974i)17-s − 19-s + (0.733 − 0.680i)23-s + (−0.988 + 0.149i)25-s + (−0.733 − 0.680i)29-s + (0.5 + 0.866i)31-s + (−0.222 + 0.974i)37-s + (0.0747 + 0.997i)41-s + (−0.0747 + 0.997i)43-s + (−0.365 − 0.930i)47-s + (−0.222 − 0.974i)53-s + (0.900 − 0.433i)55-s + ⋯
L(s)  = 1  + (0.0747 + 0.997i)5-s + (−0.365 − 0.930i)11-s + (−0.988 − 0.149i)13-s + (−0.222 + 0.974i)17-s − 19-s + (0.733 − 0.680i)23-s + (−0.988 + 0.149i)25-s + (−0.733 − 0.680i)29-s + (0.5 + 0.866i)31-s + (−0.222 + 0.974i)37-s + (0.0747 + 0.997i)41-s + (−0.0747 + 0.997i)43-s + (−0.365 − 0.930i)47-s + (−0.222 − 0.974i)53-s + (0.900 − 0.433i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.915 - 0.401i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.915 - 0.401i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.915 - 0.401i$
Analytic conductor: \(189.568\)
Root analytic conductor: \(189.568\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (463, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1764,\ (1:\ ),\ 0.915 - 0.401i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.130418158 - 0.2369131474i\)
\(L(\frac12)\) \(\approx\) \(1.130418158 - 0.2369131474i\)
\(L(1)\) \(\approx\) \(0.8770335039 + 0.1094889637i\)
\(L(1)\) \(\approx\) \(0.8770335039 + 0.1094889637i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.0747 + 0.997i)T \)
11 \( 1 + (-0.365 - 0.930i)T \)
13 \( 1 + (-0.988 - 0.149i)T \)
17 \( 1 + (-0.222 + 0.974i)T \)
19 \( 1 - T \)
23 \( 1 + (0.733 - 0.680i)T \)
29 \( 1 + (-0.733 - 0.680i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (-0.222 + 0.974i)T \)
41 \( 1 + (0.0747 + 0.997i)T \)
43 \( 1 + (-0.0747 + 0.997i)T \)
47 \( 1 + (-0.365 - 0.930i)T \)
53 \( 1 + (-0.222 - 0.974i)T \)
59 \( 1 + (-0.826 - 0.563i)T \)
61 \( 1 + (-0.733 - 0.680i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (0.222 + 0.974i)T \)
73 \( 1 + (0.623 + 0.781i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (0.988 - 0.149i)T \)
89 \( 1 + (0.623 + 0.781i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−20.10633268243335269267449289449, −19.526600593759827211321012479870, −18.662368252526090760825692058142, −17.7293034625632087225196112372, −17.12135862774162722480492322226, −16.590047508638877429369808933092, −15.54887796520327181251446869704, −15.12786331694056791557211377143, −14.08447749085622475342713326252, −13.32220543036650752074064361951, −12.52120746084150672601021636837, −12.146715332910563992787010097818, −11.10519255421348510381786614009, −10.2284867598513456318956327785, −9.2352891426337294716022435573, −9.06984387863144675996139817091, −7.68291365759707276818005228045, −7.34991961649221102355361290071, −6.188764738866609330815249879523, −5.1039378468011576601645036403, −4.76998422100867123927801679143, −3.80279566856673688524783128420, −2.4802278147974505983000218713, −1.83551527766108768631777465275, −0.58628792159709780302574676897, 0.31505089310005724377591326234, 1.767207132308378319250397214595, 2.688649960115770728728529348471, 3.338817473878261301308084692717, 4.39380256019673038834465484284, 5.33573946682778777771434432951, 6.37098259711912014356237528290, 6.73420515450982857438289651965, 7.92812441982941486027750915045, 8.396824868329725807952686802216, 9.56404454432281315801430913386, 10.33005313453185727955704671013, 10.912465581091159237122425649420, 11.59981750369321183544446018151, 12.66045636148633677455428237269, 13.27862074153640579844085119596, 14.18604984440886499873390873497, 14.911314475533829362747575752560, 15.29212526138817688976172732616, 16.424125199210467718175520182911, 17.119240469828549308851668795890, 17.769619560922927648427599864069, 18.75598014180903137198474291279, 19.12463398528010660268622324880, 19.803518205650303691241391724038

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