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Properties

Label 2-2268-252.167-c0-0-1
Degree $2$
Conductor $2268$
Sign $-0.996 - 0.0871i$
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  + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s + (−0.866 + 0.5i)7-s + (−0.707 + 0.707i)8-s + (−0.965 − 1.67i)11-s + (0.707 − 0.707i)14-s + (0.500 − 0.866i)16-s + (1.36 + 1.36i)22-s + (−0.707 + 1.22i)23-s + (0.5 + 0.866i)25-s + (−0.5 + 0.866i)28-s + (−1.22 + 0.707i)29-s + (−0.258 + 0.965i)32-s − 1.73·37-s + (−0.866 + 0.5i)43-s + (−1.67 − 0.965i)44-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s + (−0.866 + 0.5i)7-s + (−0.707 + 0.707i)8-s + (−0.965 − 1.67i)11-s + (0.707 − 0.707i)14-s + (0.500 − 0.866i)16-s + (1.36 + 1.36i)22-s + (−0.707 + 1.22i)23-s + (0.5 + 0.866i)25-s + (−0.5 + 0.866i)28-s + (−1.22 + 0.707i)29-s + (−0.258 + 0.965i)32-s − 1.73·37-s + (−0.866 + 0.5i)43-s + (−1.67 − 0.965i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.996 - 0.0871i$
Analytic conductor: \(1.13187\)
Root analytic conductor: \(1.06389\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (1511, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :0),\ -0.996 - 0.0871i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08269314399\)
\(L(\frac12)\) \(\approx\) \(0.08269314399\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
3 \( 1 \)
7 \( 1 + (0.866 - 0.5i)T \)
good5 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.965 + 1.67i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + 1.73T + T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 - 0.517iT - T^{2} \)
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 \( 1 - 0.517T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−9.383483360464791422478275246395, −8.862213263855342398733382182102, −8.158017399490682749581232895970, −7.38570460381132345887003065638, −6.57736445499189523943532971610, −5.61082237071260774012209141689, −5.43227137359261708795300643318, −3.45972398928758771373532980601, −2.96126750358873915786053515832, −1.63264614810610308897662031210, 0.07181240178406111150180165859, 1.88603586457075514061258982568, 2.67670217548251875894349887580, 3.78205977378399346414529348844, 4.70928275135958335161577009292, 5.94026394255059990284103225951, 6.86706038738275718866344333704, 7.27724944841445105950463453562, 8.107642916937101845985162983295, 8.893634377842350495872720375151

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