Properties

Label 2-3381-483.482-c0-0-0
Degree $2$
Conductor $3381$
Sign $-0.286 + 0.958i$
Analytic cond. $1.68733$
Root an. cond. $1.29897$
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  + 1.93i·2-s + (−0.991 + 0.130i)3-s − 2.73·4-s + (−0.252 − 1.91i)6-s − 3.34i·8-s + (0.965 − 0.258i)9-s + (2.70 − 0.356i)12-s + 1.58i·13-s + 3.73·16-s + (0.500 + 1.86i)18-s + i·23-s + (0.436 + 3.31i)24-s + 25-s − 3.06·26-s + (−0.923 + 0.382i)27-s + ⋯
L(s)  = 1  + 1.93i·2-s + (−0.991 + 0.130i)3-s − 2.73·4-s + (−0.252 − 1.91i)6-s − 3.34i·8-s + (0.965 − 0.258i)9-s + (2.70 − 0.356i)12-s + 1.58i·13-s + 3.73·16-s + (0.500 + 1.86i)18-s + i·23-s + (0.436 + 3.31i)24-s + 25-s − 3.06·26-s + (−0.923 + 0.382i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.286 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.286 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $-0.286 + 0.958i$
Analytic conductor: \(1.68733\)
Root analytic conductor: \(1.29897\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (3380, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3381,\ (\ :0),\ -0.286 + 0.958i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5011554375\)
\(L(\frac12)\) \(\approx\) \(0.5011554375\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.991 - 0.130i)T \)
7 \( 1 \)
23 \( 1 - iT \)
good2 \( 1 - 1.93iT - T^{2} \)
5 \( 1 - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - 1.58iT - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T^{2} \)
29 \( 1 - 1.73iT - T^{2} \)
31 \( 1 + 1.21iT - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + 1.58T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - 0.261T + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + 0.765T + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + iT - T^{2} \)
73 \( 1 - 1.98iT - T^{2} \)
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

−9.192107495538065154263865725666, −8.450814312464050700626614636087, −7.47390743254484136717023753498, −6.89312799680698996652130193297, −6.51100764873944053992121565730, −5.63091977652730628586934607293, −5.03946493053968320209674156570, −4.38053959842891395401627192191, −3.60052290901899829051335503218, −1.38924801632748305815717444346, 0.39873672058022561966972632455, 1.39410064048710565057669692772, 2.54308069623074898826216183104, 3.32740762092355845891733893782, 4.32092150534841392735775015689, 4.99727631133845639374072426711, 5.61372576838449309554344262568, 6.62253624953037042246736499100, 7.85034465761565555622428401074, 8.443508041185664964046052956947

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