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Properties

Label 2-3381-483.68-c0-0-1
Degree $2$
Conductor $3381$
Sign $-0.315 - 0.949i$
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  + (−1.67 − 0.965i)2-s + (0.608 + 0.793i)3-s + (1.36 + 2.36i)4-s + (−0.252 − 1.91i)6-s − 3.34i·8-s + (−0.258 + 0.965i)9-s + (−1.04 + 2.52i)12-s + 1.58i·13-s + (−1.86 + 3.23i)16-s + (1.36 − 1.36i)18-s + (−0.866 − 0.5i)23-s + (2.65 − 2.03i)24-s + (−0.5 − 0.866i)25-s + (1.53 − 2.65i)26-s + (−0.923 + 0.382i)27-s + ⋯
L(s)  = 1  + (−1.67 − 0.965i)2-s + (0.608 + 0.793i)3-s + (1.36 + 2.36i)4-s + (−0.252 − 1.91i)6-s − 3.34i·8-s + (−0.258 + 0.965i)9-s + (−1.04 + 2.52i)12-s + 1.58i·13-s + (−1.86 + 3.23i)16-s + (1.36 − 1.36i)18-s + (−0.866 − 0.5i)23-s + (2.65 − 2.03i)24-s + (−0.5 − 0.866i)25-s + (1.53 − 2.65i)26-s + (−0.923 + 0.382i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.315 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.315 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $-0.315 - 0.949i$
Analytic conductor: \(1.68733\)
Root analytic conductor: \(1.29897\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3381,\ (\ :0),\ -0.315 - 0.949i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4483480331\)
\(L(\frac12)\) \(\approx\) \(0.4483480331\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.608 - 0.793i)T \)
7 \( 1 \)
23 \( 1 + (0.866 + 0.5i)T \)
good2 \( 1 + (1.67 + 0.965i)T + (0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 - 1.58iT - T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 - 1.73iT - T^{2} \)
31 \( 1 + (1.05 - 0.608i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + 1.58T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.130 - 0.226i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.382 - 0.662i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + iT - T^{2} \)
73 \( 1 + (-1.71 + 0.991i)T + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
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

−9.072252224561025318232909441557, −8.625847486551876348183026157960, −7.944789078579756094613524822569, −7.13381695392675318991035215001, −6.43322967979783172782076547328, −4.89690429506441295556884320377, −3.92915394711744692362786788342, −3.34132775678542366277786112615, −2.25490855101313788393326450616, −1.67680881934069046079565362306, 0.40793094894167326830365118382, 1.61752702205422545612303219582, 2.44519466555337911381530927779, 3.64173792364425481154996932767, 5.39269240895982688592856111438, 5.83575258859717435840274833085, 6.64908453473156347033811315956, 7.39155598051545364675560018848, 7.993494508160930023089011462832, 8.226644483952044207132559321495

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