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Properties

Label 2-3381-483.482-c0-0-9
Degree $2$
Conductor $3381$
Sign $0.688 + 0.725i$
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  − 0.517i·2-s + (−0.793 − 0.608i)3-s + 0.732·4-s + (−0.315 + 0.410i)6-s − 0.896i·8-s + (0.258 + 0.965i)9-s + (−0.580 − 0.445i)12-s + 1.98i·13-s + 0.267·16-s + (0.499 − 0.133i)18-s − i·23-s + (−0.545 + 0.711i)24-s + 25-s + 1.02·26-s + (0.382 − 0.923i)27-s + ⋯
L(s)  = 1  − 0.517i·2-s + (−0.793 − 0.608i)3-s + 0.732·4-s + (−0.315 + 0.410i)6-s − 0.896i·8-s + (0.258 + 0.965i)9-s + (−0.580 − 0.445i)12-s + 1.98i·13-s + 0.267·16-s + (0.499 − 0.133i)18-s − i·23-s + (−0.545 + 0.711i)24-s + 25-s + 1.02·26-s + (0.382 − 0.923i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.688 + 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.688 + 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.688 + 0.725i$
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: \((2,\ 3381,\ (\ :0),\ 0.688 + 0.725i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.238573477\)
\(L(\frac12)\) \(\approx\) \(1.238573477\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.793 + 0.608i)T \)
7 \( 1 \)
23 \( 1 + iT \)
good2 \( 1 + 0.517iT - T^{2} \)
5 \( 1 - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - 1.98iT - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T^{2} \)
29 \( 1 - 1.73iT - T^{2} \)
31 \( 1 + 0.261iT - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - 1.98T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - 1.21T + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + 1.84T + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - iT - T^{2} \)
73 \( 1 + 1.58iT - T^{2} \)
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

−8.844204963052047219907294509189, −7.67909566860291837579547313062, −7.02810326189943296709900235662, −6.56480527980357855281634446480, −5.91303124911746665399776171357, −4.79397773979071469823212204011, −4.11255412433706860817419424344, −2.83092863563031625630096300232, −1.99939003467269022957465014669, −1.15567659767427194603405097640, 0.983545210959912465355719416376, 2.58175462689674495503495203883, 3.36788821901708467559230268002, 4.45876478020779742585787556170, 5.42320536545003180626230081539, 5.79768097644276174132267189316, 6.46015278699706677221932419894, 7.49355570991053205780930767137, 7.83973819209239134987103407160, 8.860396769998343986938230750984

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