Properties

Label 2-2268-63.34-c0-0-0
Degree $2$
Conductor $2268$
Sign $0.173 - 0.984i$
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  + (0.5 + 0.866i)7-s + (−1.5 + 0.866i)13-s + 1.73i·19-s + (−0.5 + 0.866i)25-s + 37-s + (1 − 1.73i)43-s + (−0.499 + 0.866i)49-s + (−1.5 − 0.866i)61-s + (0.5 + 0.866i)67-s + 1.73i·73-s + (0.5 − 0.866i)79-s + (−1.5 − 0.866i)91-s + (1.5 + 0.866i)97-s + (1.5 − 0.866i)103-s − 2·109-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)7-s + (−1.5 + 0.866i)13-s + 1.73i·19-s + (−0.5 + 0.866i)25-s + 37-s + (1 − 1.73i)43-s + (−0.499 + 0.866i)49-s + (−1.5 − 0.866i)61-s + (0.5 + 0.866i)67-s + 1.73i·73-s + (0.5 − 0.866i)79-s + (−1.5 − 0.866i)91-s + (1.5 + 0.866i)97-s + (1.5 − 0.866i)103-s − 2·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.173 - 0.984i$
Analytic conductor: \(1.13187\)
Root analytic conductor: \(1.06389\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (1189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :0),\ 0.173 - 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.047281592\)
\(L(\frac12)\) \(\approx\) \(1.047281592\)
\(L(1)\) 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 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - 1.73iT - T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - T + T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - 1.73iT - T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
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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

−9.420056535421049446472641110578, −8.642851317633414069579636949229, −7.78153581652768383801341344239, −7.23799822220650440798619854488, −6.12118354656560318744628813082, −5.46934323291334221500532674465, −4.64567970474956593744742090617, −3.72391015436283279827454019357, −2.47442362313862663710996778273, −1.73068103862767334495426726424, 0.71197459327781752028770392355, 2.28481665526116421541602402999, 3.12270504489969288711812826615, 4.51977472756465596439625184697, 4.75202940456689191078574519267, 5.90758592807487326334319707482, 6.85433948495754649695117689604, 7.61235372126608453422466007365, 8.006807662205299319200684831470, 9.165248785724656983582651011530

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