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Properties

Label 2-3381-483.68-c0-0-21
Degree $2$
Conductor $3381$
Sign $0.469 - 0.883i$
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 + (0.793 − 0.608i)3-s + (1.36 + 2.36i)4-s + (1.91 − 0.252i)6-s + 3.34i·8-s + (0.258 − 0.965i)9-s + (2.52 + 1.04i)12-s − 1.21i·13-s + (−1.86 + 3.23i)16-s + (1.36 − 1.36i)18-s + (−0.866 − 0.5i)23-s + (2.03 + 2.65i)24-s + (−0.5 − 0.866i)25-s + (1.17 − 2.03i)26-s + (−0.382 − 0.923i)27-s + ⋯
L(s)  = 1  + (1.67 + 0.965i)2-s + (0.793 − 0.608i)3-s + (1.36 + 2.36i)4-s + (1.91 − 0.252i)6-s + 3.34i·8-s + (0.258 − 0.965i)9-s + (2.52 + 1.04i)12-s − 1.21i·13-s + (−1.86 + 3.23i)16-s + (1.36 − 1.36i)18-s + (−0.866 − 0.5i)23-s + (2.03 + 2.65i)24-s + (−0.5 − 0.866i)25-s + (1.17 − 2.03i)26-s + (−0.382 − 0.923i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.469 - 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.469 - 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.469 - 0.883i$
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: \((2,\ 3381,\ (\ :0),\ 0.469 - 0.883i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.182631707\)
\(L(\frac12)\) \(\approx\) \(4.182631707\)
\(L(1)\) 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 \( 1 + (0.866 + 0.5i)T \)
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 \( 1 + 1.21iT - T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 - 1.73iT - T^{2} \)
31 \( 1 + (1.37 - 0.793i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 - 1.21T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.991 - 1.71i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.923 + 1.60i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + iT - T^{2} \)
73 \( 1 + (0.226 - 0.130i)T + (0.5 - 0.866i)T^{2} \)
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

−8.431452048275486866146792205688, −7.87392532838811042753612581076, −7.38524461401665221573403180269, −6.53456988699867199230302213997, −6.00370722538824329199386148848, −5.17468526801896343970020667512, −4.32318371175372129779559923448, −3.41754217141728742528779267677, −2.93897122954316230694504914716, −1.88557639177924695893229965469, 1.76951950807132564726971075268, 2.26424228880948716699911164770, 3.31221192538548455919128050881, 4.07481730245945605532760821122, 4.35519259744885891759145950144, 5.44321147739319740160813389806, 5.97749894174989069986353616052, 7.02999174905021370048650173081, 7.77677604646863842994275431704, 9.032801260422988045019113804571

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