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Properties

Label 2-3276-91.38-c0-0-0
Degree $2$
Conductor $3276$
Sign $0.386 - 0.922i$
Analytic cond. $1.63493$
Root an. cond. $1.27864$
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  + (0.5 + 0.866i)7-s + (−0.5 + 0.866i)13-s + (0.5 + 0.866i)19-s + (0.5 − 0.866i)25-s + (−0.5 + 0.866i)31-s + (−1.5 + 0.866i)37-s + 43-s + (−0.499 + 0.866i)49-s + (1.5 + 0.866i)67-s + (−0.5 + 0.866i)73-s + (−0.5 − 0.866i)79-s − 0.999·91-s + 2·97-s + (1.5 − 0.866i)103-s + (−1.5 − 0.866i)109-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)7-s + (−0.5 + 0.866i)13-s + (0.5 + 0.866i)19-s + (0.5 − 0.866i)25-s + (−0.5 + 0.866i)31-s + (−1.5 + 0.866i)37-s + 43-s + (−0.499 + 0.866i)49-s + (1.5 + 0.866i)67-s + (−0.5 + 0.866i)73-s + (−0.5 − 0.866i)79-s − 0.999·91-s + 2·97-s + (1.5 − 0.866i)103-s + (−1.5 − 0.866i)109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3276\)    =    \(2^{2} \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.386 - 0.922i$
Analytic conductor: \(1.63493\)
Root analytic conductor: \(1.27864\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3276} (1585, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3276,\ (\ :0),\ 0.386 - 0.922i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.229719182\)
\(L(\frac12)\) \(\approx\) \(1.229719182\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T^{2} \)
97 \( 1 - 2T + T^{2} \)
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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

−8.808018708222388361083739690713, −8.401355427901258069745534586826, −7.45735289571565894306087454559, −6.75956202037514628747692063659, −5.88678262867663424321559775989, −5.15096420313049167300489309310, −4.44807774789521778769943661440, −3.39270829817011179627149256740, −2.39169030920415275690909324094, −1.51273442492794027975358946607, 0.76928232423461740911054470868, 2.05766945114930847803090718769, 3.16915981514255235502836625509, 3.97272916804872554483507145888, 4.97000157823228307008810694099, 5.44319723677949011773746802858, 6.56800741298661335828446199808, 7.43222892996541527102605340630, 7.66521178533855076083032342665, 8.721428239091977409328540396038

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