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Properties

Label 2-2268-63.11-c0-0-0
Degree $2$
Conductor $2268$
Sign $0.959 - 0.281i$
Analytic cond. $1.13187$
Root an. cond. $1.06389$
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  + 7-s + (0.5 + 0.866i)13-s + (0.5 + 0.866i)19-s + (−0.5 − 0.866i)25-s − 31-s + (0.5 + 0.866i)37-s + (0.5 − 0.866i)43-s + 49-s + 2·61-s − 67-s + (0.5 − 0.866i)73-s − 79-s + (0.5 + 0.866i)91-s + (−1 + 1.73i)97-s + (0.5 − 0.866i)103-s + ⋯
L(s)  = 1  + 7-s + (0.5 + 0.866i)13-s + (0.5 + 0.866i)19-s + (−0.5 − 0.866i)25-s − 31-s + (0.5 + 0.866i)37-s + (0.5 − 0.866i)43-s + 49-s + 2·61-s − 67-s + (0.5 − 0.866i)73-s − 79-s + (0.5 + 0.866i)91-s + (−1 + 1.73i)97-s + (0.5 − 0.866i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 - 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 - 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.959 - 0.281i$
Analytic conductor: \(1.13187\)
Root analytic conductor: \(1.06389\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (1565, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :0),\ 0.959 - 0.281i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.358473425\)
\(L(\frac12)\) \(\approx\) \(1.358473425\)
\(L(1)\) 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 \( 1 - T \)
good5 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 2T + T^{2} \)
67 \( 1 + T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + T + T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
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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

−9.153338232986323186681088460622, −8.429828042820330762870312692366, −7.79530403988971392926624644083, −6.99001581039198255487883852803, −6.06049256396177347611625316284, −5.30844406603584785401115148180, −4.36660985697126908925200191638, −3.67283423915175459448753760639, −2.30551555332520783623695991816, −1.38506309809481737495560227578, 1.14599634746030754773680385690, 2.34228919283796378074577378497, 3.43488079466704275041015761409, 4.37248025714459566018551953218, 5.32608295124416406390418438935, 5.81569501731695088579930511292, 7.05323076797002470098049645351, 7.64073232880201584420174544260, 8.376110809271684023258551777402, 9.108688689345120411643750672502

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