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Properties

Label 2-2268-63.20-c1-0-27
Degree $2$
Conductor $2268$
Sign $0.0155 + 0.999i$
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 + (−1.5 + 0.866i)13-s − 8.66i·19-s + (2.5 − 4.33i)25-s + (−9 + 5.19i)31-s + 37-s + (4 − 6.92i)43-s + (1.00 − 6.92i)49-s + (7.5 + 4.33i)61-s + (−5.5 − 9.52i)67-s + 1.73i·73-s + (6.5 − 11.2i)79-s + (−1.50 + 4.33i)91-s + (−16.5 − 9.52i)97-s + (−16.5 + 9.52i)103-s + ⋯
L(s)  = 1  + (0.755 − 0.654i)7-s + (−0.416 + 0.240i)13-s − 1.98i·19-s + (0.5 − 0.866i)25-s + (−1.61 + 0.933i)31-s + 0.164·37-s + (0.609 − 1.05i)43-s + (0.142 − 0.989i)49-s + (0.960 + 0.554i)61-s + (−0.671 − 1.16i)67-s + 0.202i·73-s + (0.731 − 1.26i)79-s + (−0.157 + 0.453i)91-s + (−1.67 − 0.967i)97-s + (−1.62 + 0.938i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0155 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0155 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.0155 + 0.999i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (1889, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ 0.0155 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.559651328\)
\(L(\frac12)\) \(\approx\) \(1.559651328\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.5 - 0.866i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 8.66iT - 19T^{2} \)
23 \( 1 + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (9 - 5.19i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - T + 37T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4 + 6.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.5 - 4.33i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.5 + 9.52i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 1.73iT - 73T^{2} \)
79 \( 1 + (-6.5 + 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (16.5 + 9.52i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.876609393762842801708941162212, −8.058591577841296844896059246569, −7.12119035389214165640408718479, −6.82824316388377831053822748708, −5.50574558097363266176893818596, −4.79553712772513149935643037734, −4.09962165014308593599537602616, −2.92199407609548793006755924537, −1.88349680519783998838227727060, −0.54459385875799486262768981193, 1.40258669156789020295432180216, 2.33958643739715296745029178274, 3.49081813135816885939045537244, 4.39482547467703296118627195461, 5.48134602866110470966231110422, 5.78818784945066624681164625895, 6.99701297045068367672881413887, 7.83403709587226284804162897743, 8.290516287761940316948285894917, 9.271998846550103101975878626562

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