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Properties

Label 2-2268-21.5-c1-0-4
Degree $2$
Conductor $2268$
Sign $-0.407 + 0.913i$
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  + (−2.11 + 3.67i)5-s + (−2.57 + 0.604i)7-s + (−4.69 + 2.70i)11-s + 4.28i·13-s + (3.26 + 5.66i)17-s + (−0.653 − 0.377i)19-s + (−3.22 − 1.86i)23-s + (−6.48 − 11.2i)25-s + 4.09i·29-s + (6.17 − 3.56i)31-s + (3.24 − 10.7i)35-s + (−2.90 + 5.02i)37-s + 1.41·41-s + 3.23·43-s + (−2.06 + 3.57i)47-s + ⋯
L(s)  = 1  + (−0.947 + 1.64i)5-s + (−0.973 + 0.228i)7-s + (−1.41 + 0.816i)11-s + 1.18i·13-s + (0.792 + 1.37i)17-s + (−0.149 − 0.0865i)19-s + (−0.672 − 0.388i)23-s + (−1.29 − 2.24i)25-s + 0.760i·29-s + (1.10 − 0.640i)31-s + (0.547 − 1.81i)35-s + (−0.477 + 0.826i)37-s + 0.220·41-s + 0.493·43-s + (−0.301 + 0.521i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.407 + 0.913i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.407 + 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.407 + 0.913i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (2105, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ -0.407 + 0.913i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4573543235\)
\(L(\frac12)\) \(\approx\) \(0.4573543235\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.57 - 0.604i)T \)
good5 \( 1 + (2.11 - 3.67i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (4.69 - 2.70i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 4.28iT - 13T^{2} \)
17 \( 1 + (-3.26 - 5.66i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.653 + 0.377i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.22 + 1.86i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 4.09iT - 29T^{2} \)
31 \( 1 + (-6.17 + 3.56i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.90 - 5.02i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 1.41T + 41T^{2} \)
43 \( 1 - 3.23T + 43T^{2} \)
47 \( 1 + (2.06 - 3.57i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.34 + 3.08i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.610 - 1.05i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.08 - 1.77i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.22 + 12.5i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.83iT - 71T^{2} \)
73 \( 1 + (8.39 - 4.84i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.51 - 4.35i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 0.771T + 83T^{2} \)
89 \( 1 + (-7.04 + 12.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.12iT - 97T^{2} \)
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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

−9.926464489616071685863046450348, −8.621099277829701819597050176684, −7.84120629557644922930929930124, −7.22634345838632954428399005109, −6.51874550081967822781399483713, −5.93268657435704687166614062088, −4.54357733144079397933302714703, −3.74491711320682005553860645858, −2.93255776147166431759302605779, −2.16423569413886984887255793267, 0.21360356536369298531772298379, 0.828305735563600274232633327477, 2.77153225286384546177991310346, 3.51196174654907561521568236563, 4.49854368271836568109085956978, 5.41937687857885428736670329111, 5.75394944926275407044010249969, 7.26061055718890583410235494470, 7.86239476669364928793978790636, 8.365175216607274631071903624950

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