Properties

Label 2-3775-151.150-c0-0-9
Degree $2$
Conductor $3775$
Sign $-i$
Analytic cond. $1.88397$
Root an. cond. $1.37257$
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  − 1.61·2-s + i·3-s + 1.61·4-s − 1.61i·6-s + 1.61i·7-s − 8-s + 1.61i·12-s − 0.618i·13-s − 2.61i·14-s + 17-s + 19-s − 1.61·21-s − i·23-s − i·24-s + 1.00i·26-s + i·27-s + ⋯
L(s)  = 1  − 1.61·2-s + i·3-s + 1.61·4-s − 1.61i·6-s + 1.61i·7-s − 8-s + 1.61i·12-s − 0.618i·13-s − 2.61i·14-s + 17-s + 19-s − 1.61·21-s − i·23-s − i·24-s + 1.00i·26-s + i·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3775\)    =    \(5^{2} \cdot 151\)
Sign: $-i$
Analytic conductor: \(1.88397\)
Root analytic conductor: \(1.37257\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3775} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3775,\ (\ :0),\ -i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6899882232\)
\(L(\frac12)\) \(\approx\) \(0.6899882232\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
151 \( 1 + iT \)
good2 \( 1 + 1.61T + T^{2} \)
3 \( 1 - iT - T^{2} \)
7 \( 1 - 1.61iT - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + 0.618iT - T^{2} \)
17 \( 1 - T + T^{2} \)
19 \( 1 - T + T^{2} \)
23 \( 1 + iT - T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - 1.61T + T^{2} \)
37 \( 1 - 0.618T + T^{2} \)
41 \( 1 + 0.618iT - T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 - 0.618T + T^{2} \)
53 \( 1 + 0.618iT - T^{2} \)
59 \( 1 - T + T^{2} \)
61 \( 1 - 1.61iT - T^{2} \)
67 \( 1 - iT - T^{2} \)
71 \( 1 + iT - T^{2} \)
73 \( 1 + 0.618iT - T^{2} \)
79 \( 1 + 0.618iT - T^{2} \)
83 \( 1 + 1.61iT - T^{2} \)
89 \( 1 - 1.61iT - T^{2} \)
97 \( 1 + 1.61T + T^{2} \)
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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

−9.076295961532517132243812751202, −8.318395008492499500385113951417, −7.78983331566939419699951837331, −6.87092158529079034198830087464, −5.87852590720914321228298207629, −5.29229454456766846511699080650, −4.32401682114917741572327239534, −3.04722268168267606122927950236, −2.42704887086695396275974291578, −1.09232838475925318222751629101, 1.03012528293985167961991197646, 1.22207385737265184191004880126, 2.49781002083005816667273398793, 3.70709775251671907645112967218, 4.64806617430952268754250124934, 5.97339702749532785938744354231, 6.87730757471352394578527240049, 7.17620905531179257478888693692, 7.87267882255148109468658851206, 8.127699393116143167960914054336

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