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Properties

Label 2-3381-483.206-c0-0-18
Degree $2$
Conductor $3381$
Sign $0.704 + 0.709i$
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  + (1.22 − 0.707i)2-s + (0.991 + 0.130i)3-s + (0.499 − 0.866i)4-s + (1.30 − 0.541i)6-s + (0.965 + 0.258i)9-s + (0.608 − 0.793i)12-s − 1.84i·13-s + (0.499 + 0.866i)16-s + (1.36 − 0.366i)18-s + (−0.866 + 0.5i)23-s + (−0.5 + 0.866i)25-s + (−1.30 − 2.26i)26-s + (0.923 + 0.382i)27-s + (0.662 + 0.382i)31-s + (1.22 + 0.707i)32-s + ⋯
L(s)  = 1  + (1.22 − 0.707i)2-s + (0.991 + 0.130i)3-s + (0.499 − 0.866i)4-s + (1.30 − 0.541i)6-s + (0.965 + 0.258i)9-s + (0.608 − 0.793i)12-s − 1.84i·13-s + (0.499 + 0.866i)16-s + (1.36 − 0.366i)18-s + (−0.866 + 0.5i)23-s + (−0.5 + 0.866i)25-s + (−1.30 − 2.26i)26-s + (0.923 + 0.382i)27-s + (0.662 + 0.382i)31-s + (1.22 + 0.707i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.704 + 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.704 + 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.704 + 0.709i$
Analytic conductor: \(1.68733\)
Root analytic conductor: \(1.29897\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (2138, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3381,\ (\ :0),\ 0.704 + 0.709i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.395619825\)
\(L(\frac12)\) \(\approx\) \(3.395619825\)
\(L(1)\) 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 \( 1 + (0.866 - 0.5i)T \)
good2 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + 1.84iT - T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-0.662 - 0.382i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + 1.84T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.923 + 1.60i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + 2iT - T^{2} \)
73 \( 1 + (-0.662 - 0.382i)T + (0.5 + 0.866i)T^{2} \)
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

−8.495943207490952862170475493441, −8.107089607020432024831945428923, −7.31426690657573775314322487041, −6.20706673037693897349361568539, −5.31732282749135173542091977371, −4.80617763046787447149480008001, −3.58810712090971204436154849578, −3.42822397142926391550633579790, −2.47547661384824010795032029987, −1.55620400877782436550130130242, 1.67824798707490200027239994789, 2.65012577284091054862302806852, 3.67106280487782297025442456011, 4.31255474690873046066890676026, 4.79902863517528947394255872272, 6.06846057754627921682852205911, 6.59110882686670185389932707213, 7.16093512419769177141286362849, 8.064582606119866957618089842480, 8.622781607307026595408320049672

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