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Properties

Label 2-3267-99.79-c0-0-1
Degree $2$
Conductor $3267$
Sign $0.822 - 0.568i$
Analytic cond. $1.63044$
Root an. cond. $1.27688$
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  + (−0.104 + 0.994i)4-s + (0.669 − 0.743i)5-s + (−0.978 − 0.207i)16-s + (0.669 + 0.743i)20-s + (1 + 1.73i)23-s + (0.978 − 0.207i)31-s + (0.809 + 0.587i)37-s + (−0.104 − 0.994i)47-s + (0.669 − 0.743i)49-s + (0.309 − 0.951i)53-s + (−0.104 + 0.994i)59-s + (0.309 − 0.951i)64-s + (0.5 + 0.866i)67-s + (0.309 + 0.951i)71-s + (−0.809 + 0.587i)80-s + ⋯
L(s)  = 1  + (−0.104 + 0.994i)4-s + (0.669 − 0.743i)5-s + (−0.978 − 0.207i)16-s + (0.669 + 0.743i)20-s + (1 + 1.73i)23-s + (0.978 − 0.207i)31-s + (0.809 + 0.587i)37-s + (−0.104 − 0.994i)47-s + (0.669 − 0.743i)49-s + (0.309 − 0.951i)53-s + (−0.104 + 0.994i)59-s + (0.309 − 0.951i)64-s + (0.5 + 0.866i)67-s + (0.309 + 0.951i)71-s + (−0.809 + 0.587i)80-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.822 - 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.822 - 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3267\)    =    \(3^{3} \cdot 11^{2}\)
Sign: $0.822 - 0.568i$
Analytic conductor: \(1.63044\)
Root analytic conductor: \(1.27688\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3267} (3016, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3267,\ (\ :0),\ 0.822 - 0.568i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.401574510\)
\(L(\frac12)\) \(\approx\) \(1.401574510\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.104 - 0.994i)T^{2} \)
5 \( 1 + (-0.669 + 0.743i)T + (-0.104 - 0.994i)T^{2} \)
7 \( 1 + (-0.669 + 0.743i)T^{2} \)
13 \( 1 + (-0.913 + 0.406i)T^{2} \)
17 \( 1 + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (-0.309 + 0.951i)T^{2} \)
23 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.669 + 0.743i)T^{2} \)
31 \( 1 + (-0.978 + 0.207i)T + (0.913 - 0.406i)T^{2} \)
37 \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \)
41 \( 1 + (-0.669 - 0.743i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.104 + 0.994i)T + (-0.978 + 0.207i)T^{2} \)
53 \( 1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2} \)
59 \( 1 + (0.104 - 0.994i)T + (-0.978 - 0.207i)T^{2} \)
61 \( 1 + (-0.913 - 0.406i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2} \)
73 \( 1 + (-0.309 - 0.951i)T^{2} \)
79 \( 1 + (0.104 - 0.994i)T^{2} \)
83 \( 1 + (-0.913 - 0.406i)T^{2} \)
89 \( 1 + 2T + T^{2} \)
97 \( 1 + (0.669 + 0.743i)T + (-0.104 + 0.994i)T^{2} \)
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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

−8.769539150017876140281597480399, −8.330737449315502044503137536568, −7.40118935745164603358864984567, −6.83591432990027421748733340296, −5.72051340553335638816133807860, −5.11112400706558713659921782103, −4.23243117792972020342447232209, −3.36781643969641202045126232185, −2.44720772116325749108722256702, −1.26888333004811860866723755382, 0.994947766909477974499249911054, 2.26549419501268866170171849637, 2.88127019819479572893797242291, 4.29619235794349716629726656101, 4.94705350169463961182600160786, 5.92521305038675241594687941927, 6.40275797762159290390327585096, 6.99796025403169469286419601358, 8.051812414919073347348444844306, 8.946658968489712595620682824372

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