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Properties

Label 2-3267-99.79-c0-0-0
Degree $2$
Conductor $3267$
Sign $-0.149 - 0.988i$
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  + (−1.05 + 0.946i)2-s + (0.104 − 0.994i)4-s + (0.669 − 0.743i)5-s + (0.575 + 1.29i)7-s + 1.41i·10-s + (−1.82 − 0.813i)14-s + (0.978 + 0.207i)16-s + (−0.669 − 0.743i)20-s + (1.34 − 0.437i)28-s + (0.575 + 1.29i)29-s + (−0.978 + 0.207i)31-s + (−1.22 + 0.707i)32-s + (1.34 + 0.437i)35-s + (0.809 + 0.587i)37-s + (−0.104 − 0.994i)47-s + ⋯
L(s)  = 1  + (−1.05 + 0.946i)2-s + (0.104 − 0.994i)4-s + (0.669 − 0.743i)5-s + (0.575 + 1.29i)7-s + 1.41i·10-s + (−1.82 − 0.813i)14-s + (0.978 + 0.207i)16-s + (−0.669 − 0.743i)20-s + (1.34 − 0.437i)28-s + (0.575 + 1.29i)29-s + (−0.978 + 0.207i)31-s + (−1.22 + 0.707i)32-s + (1.34 + 0.437i)35-s + (0.809 + 0.587i)37-s + (−0.104 − 0.994i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.149 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.149 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3267\)    =    \(3^{3} \cdot 11^{2}\)
Sign: $-0.149 - 0.988i$
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: \((2,\ 3267,\ (\ :0),\ -0.149 - 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8385158395\)
\(L(\frac12)\) \(\approx\) \(0.8385158395\)
\(L(1)\) 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 \( 1 + (1.05 - 0.946i)T + (0.104 - 0.994i)T^{2} \)
5 \( 1 + (-0.669 + 0.743i)T + (-0.104 - 0.994i)T^{2} \)
7 \( 1 + (-0.575 - 1.29i)T + (-0.669 + 0.743i)T^{2} \)
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 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.575 - 1.29i)T + (-0.669 + 0.743i)T^{2} \)
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 \( 1 + (-0.294 + 1.38i)T + (-0.913 - 0.406i)T^{2} \)
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 \( 1 + (-0.831 + 1.14i)T + (-0.309 - 0.951i)T^{2} \)
79 \( 1 + (0.104 - 0.994i)T^{2} \)
83 \( 1 + (0.294 - 1.38i)T + (-0.913 - 0.406i)T^{2} \)
89 \( 1 + T^{2} \)
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

−8.875209281309099304413383225257, −8.456575489219135199258142768811, −7.75143436673168176759806799549, −6.84353609735706319822650591388, −6.11260387944916303676526505838, −5.37012186674422056324369935003, −4.93590701241613357464966347281, −3.45038715937489672898463493549, −2.19209606933399459689158884847, −1.23867089066437498108644143062, 0.821829644646311025597775298674, 1.90015704873942300471707223656, 2.65020510312082488044075245792, 3.67247545829708983302722207434, 4.55768785846129787443908796992, 5.71748318297454160358940938034, 6.49796580862556547508359017279, 7.46971045709272223816740476681, 7.889996433344560142141658467181, 8.808625685508794493427708559991

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