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Properties

Label 2-3381-483.482-c0-0-18
Degree $2$
Conductor $3381$
Sign $0.284 - 0.958i$
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 + (−0.130 + 0.991i)3-s − 2.73·4-s + (−1.91 − 0.252i)6-s − 3.34i·8-s + (−0.965 − 0.258i)9-s + (0.356 − 2.70i)12-s − 1.21i·13-s + 3.73·16-s + (0.500 − 1.86i)18-s − i·23-s + (3.31 + 0.436i)24-s + 25-s + 2.35·26-s + (0.382 − 0.923i)27-s + ⋯
L(s)  = 1  + 1.93i·2-s + (−0.130 + 0.991i)3-s − 2.73·4-s + (−1.91 − 0.252i)6-s − 3.34i·8-s + (−0.965 − 0.258i)9-s + (0.356 − 2.70i)12-s − 1.21i·13-s + 3.73·16-s + (0.500 − 1.86i)18-s − i·23-s + (3.31 + 0.436i)24-s + 25-s + 2.35·26-s + (0.382 − 0.923i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.284 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.284 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.284 - 0.958i$
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: \((2,\ 3381,\ (\ :0),\ 0.284 - 0.958i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6128465832\)
\(L(\frac12)\) \(\approx\) \(0.6128465832\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.130 - 0.991i)T \)
7 \( 1 \)
23 \( 1 + iT \)
good2 \( 1 - 1.93iT - T^{2} \)
5 \( 1 - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + 1.21iT - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T^{2} \)
29 \( 1 + 1.73iT - T^{2} \)
31 \( 1 + 1.58iT - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + 1.21T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + 1.98T + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + 1.84T + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - iT - T^{2} \)
73 \( 1 + 0.261iT - T^{2} \)
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

−8.635692676802827437576912756702, −8.180323038668997826223717876900, −7.56321705868355120962461896460, −6.44170257654180911443366787620, −6.06634153757711587812797621592, −5.21729130816361929143119419277, −4.67256476050989268301361419346, −3.91975179423421108769954375389, −2.94135892271173649187717571393, −0.39898775858521285646931531508, 1.40680070377443513682352996509, 1.74700802678756720163370621001, 3.00452029930528994196081307141, 3.46261616215112923694015869615, 4.78296670889289163900081012240, 5.17110607604697176504669962060, 6.43367491493938202036862435766, 7.20946760553984570460935963883, 8.256162304752026945469135998566, 8.876407303732556436621530347670

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