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Properties

Label 2-3800-760.189-c0-0-3
Degree $2$
Conductor $3800$
Sign $0.447 - 0.894i$
Analytic cond. $1.89644$
Root an. cond. $1.37711$
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  − i·2-s + 1.53i·3-s − 4-s + 1.53·6-s − 0.347i·7-s + i·8-s − 1.34·9-s − 1.53i·12-s − 1.87i·13-s − 0.347·14-s + 16-s + 1.87i·17-s + 1.34i·18-s − 19-s + 0.532·21-s + ⋯
L(s)  = 1  − i·2-s + 1.53i·3-s − 4-s + 1.53·6-s − 0.347i·7-s + i·8-s − 1.34·9-s − 1.53i·12-s − 1.87i·13-s − 0.347·14-s + 16-s + 1.87i·17-s + 1.34i·18-s − 19-s + 0.532·21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3800\)    =    \(2^{3} \cdot 5^{2} \cdot 19\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(1.89644\)
Root analytic conductor: \(1.37711\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3800} (949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3800,\ (\ :0),\ 0.447 - 0.894i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9495755422\)
\(L(\frac12)\) \(\approx\) \(0.9495755422\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
5 \( 1 \)
19 \( 1 + T \)
good3 \( 1 - 1.53iT - T^{2} \)
7 \( 1 + 0.347iT - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + 1.87iT - T^{2} \)
17 \( 1 - 1.87iT - T^{2} \)
23 \( 1 - 1.53iT - T^{2} \)
29 \( 1 - 1.87T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - iT - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - iT - T^{2} \)
53 \( 1 - 0.347iT - T^{2} \)
59 \( 1 + 1.53T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + 0.347iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 1.53iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.954768252258581465507969654458, −8.317592686635888725968014212833, −7.80599719833522009170729227285, −6.21069422207521111440581695158, −5.54215302171513635130467628941, −4.75948594425292033401796425479, −4.08343082045360354246643870477, −3.43340084325485757930990958980, −2.76541946939768003642827771560, −1.29774912106516932850348464564, 0.59391308421665691461222734084, 1.95578864965617538534759869238, 2.78718196518846251748011518083, 4.30327564757625793169181641224, 4.83688218040819580912583476817, 5.96098234393175694228500674720, 6.64698682686890215100202189473, 6.88176350504426104081348863350, 7.58613008060066245343639524592, 8.521596461789546927598301523378

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