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Properties

Label 2-3381-483.482-c0-0-11
Degree $2$
Conductor $3381$
Sign $-0.524 + 0.851i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.524 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.524 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $-0.524 + 0.851i$
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: \((2,\ 3381,\ (\ :0),\ -0.524 + 0.851i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.546283109\)
\(L(\frac12)\) \(\approx\) \(1.546283109\)
\(L(1)\) 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

−8.795384604321177414633851622700, −8.302641066188239800897037517047, −7.35633868543091291620379022249, −6.20355620892101780336848062233, −4.88163144503689458510659496702, −4.20212901560750227993185537294, −3.83153898165373243714135649361, −2.50477945944693817696857969429, −2.34901339662605954371741189102, −1.11364414852585330581201139166, 1.14332418261764267744160213262, 3.05527599492178194156696758431, 3.65301390484059867695949338255, 4.80662251961034520481633522556, 5.30357557047670579788491872074, 6.23013622966414868435083326371, 7.00649420517502556133032890740, 7.59283081257570061829359200500, 8.060563523486235158972401712909, 8.897251013831066322212711164280

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