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Properties

Label 2-2268-63.16-c1-0-20
Degree $2$
Conductor $2268$
Sign $0.0788 + 0.996i$
Analytic cond. $18.1100$
Root an. cond. $4.25559$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2 + 1.73i)7-s + (−1 − 1.73i)13-s + (0.5 − 0.866i)19-s − 5·25-s + (3.5 − 6.06i)31-s + (5 − 8.66i)37-s + (−2.5 + 4.33i)43-s + (1.00 − 6.92i)49-s + (0.5 + 0.866i)61-s + (8 − 13.8i)67-s + (−8.5 − 14.7i)73-s + (2 + 3.46i)79-s + (5 + 1.73i)91-s + (9.5 − 16.4i)97-s + 20·103-s + ⋯
L(s)  = 1  + (−0.755 + 0.654i)7-s + (−0.277 − 0.480i)13-s + (0.114 − 0.198i)19-s − 25-s + (0.628 − 1.08i)31-s + (0.821 − 1.42i)37-s + (−0.381 + 0.660i)43-s + (0.142 − 0.989i)49-s + (0.0640 + 0.110i)61-s + (0.977 − 1.69i)67-s + (−0.994 − 1.72i)73-s + (0.225 + 0.389i)79-s + (0.524 + 0.181i)91-s + (0.964 − 1.67i)97-s + 1.97·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0788 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0788 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.0788 + 0.996i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ 0.0788 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.007679373\)
\(L(\frac12)\) \(\approx\) \(1.007679373\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 - 1.73i)T \)
good5 \( 1 + 5T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + (1 + 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.5 + 6.06i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5 + 8.66i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.5 - 4.33i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-8 + 13.8i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (8.5 + 14.7i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-9.5 + 16.4i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.958011323934146674340728859508, −8.007623087298332746234245775671, −7.40088258895552779700720193226, −6.30817956212715710106039443967, −5.85281977791787056277457947149, −4.89847339124623733294036718634, −3.87201055401858139275699630868, −2.93978152154037080513733695321, −2.06653202313522153868653093689, −0.37259392282041021892127344074, 1.16353929674270346245792418559, 2.51000669199540901406570886824, 3.51342737769171320840106795590, 4.28581574043032567823460635352, 5.22556683155357210348381521104, 6.25246162473715572955443888923, 6.82927548833839094404464875102, 7.61218400205677995047146541298, 8.425267446671817411179928971512, 9.291122328611718349895599706417

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