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Properties

Label 2-3381-483.482-c0-0-6
Degree $2$
Conductor $3381$
Sign $-0.0263 - 0.999i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0263 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0263 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $-0.0263 - 0.999i$
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: \((2,\ 3381,\ (\ :0),\ -0.0263 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.012365224\)
\(L(\frac12)\) \(\approx\) \(1.012365224\)
\(L(1)\) 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

−8.672403289575849515315401300949, −8.128861959735602818013175847238, −7.20892143185610051885318669991, −6.90130014918841567210427859868, −6.21658206595271703251103326005, −5.56715580799837483334529095828, −4.62692108245661745278458460193, −3.90761406261636248839214603049, −2.50360998470015906985301341229, −0.914332594309892729873929502040, 0.878021660178044648186178728733, 2.29444502562938157146106872895, 3.04036965485628837794889020283, 3.67505558397674947101678970332, 4.42454628731002370923215939303, 5.27731546902533290225687210558, 5.74091505714260272056612220663, 7.49003499785018473807667996056, 8.340756410962237783134415127527, 8.982112559951128386582556353480

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