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Properties

Label 2-2268-21.17-c1-0-25
Degree $2$
Conductor $2268$
Sign $-0.303 + 0.952i$
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  + (0.0382 + 0.0661i)5-s + (2.16 − 1.51i)7-s + (−4.66 − 2.69i)11-s + 5.31i·13-s + (1.89 − 3.27i)17-s + (−4.33 + 2.50i)19-s + (2.02 − 1.16i)23-s + (2.49 − 4.32i)25-s − 10.2i·29-s + (−4.97 − 2.87i)31-s + (0.183 + 0.0852i)35-s + (0.354 + 0.613i)37-s + 6.59·41-s − 1.43·43-s + (1.46 + 2.53i)47-s + ⋯
L(s)  = 1  + (0.0170 + 0.0295i)5-s + (0.818 − 0.574i)7-s + (−1.40 − 0.811i)11-s + 1.47i·13-s + (0.458 − 0.794i)17-s + (−0.995 + 0.574i)19-s + (0.422 − 0.243i)23-s + (0.499 − 0.865i)25-s − 1.89i·29-s + (−0.893 − 0.516i)31-s + (0.0309 + 0.0144i)35-s + (0.0582 + 0.100i)37-s + 1.03·41-s − 0.218·43-s + (0.213 + 0.369i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.303 + 0.952i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.303 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.303 + 0.952i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (1781, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ -0.303 + 0.952i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.212417682\)
\(L(\frac12)\) \(\approx\) \(1.212417682\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.16 + 1.51i)T \)
good5 \( 1 + (-0.0382 - 0.0661i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (4.66 + 2.69i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 5.31iT - 13T^{2} \)
17 \( 1 + (-1.89 + 3.27i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.33 - 2.50i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.02 + 1.16i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 10.2iT - 29T^{2} \)
31 \( 1 + (4.97 + 2.87i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.354 - 0.613i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 6.59T + 41T^{2} \)
43 \( 1 + 1.43T + 43T^{2} \)
47 \( 1 + (-1.46 - 2.53i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (10.4 + 6.05i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.289 + 0.502i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.40 + 1.38i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.63 - 4.56i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.32iT - 71T^{2} \)
73 \( 1 + (6.17 + 3.56i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.469 + 0.812i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 12.9T + 83T^{2} \)
89 \( 1 + (-1.51 - 2.62i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 7.13iT - 97T^{2} \)
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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

−8.636060123650590785858572234635, −8.013190746052146844019167285978, −7.40242027134712626714235058713, −6.43770535515147769782488164858, −5.63775322756763114034037362957, −4.65589687636887123796425540762, −4.10695163591380395285041634896, −2.80191722068047936711725469171, −1.89660729710654213747643275727, −0.40850208710211052638862502560, 1.42019594488899889620249561108, 2.51839998375635485339673024478, 3.34448547918381912711962227199, 4.73964590265112188985845242457, 5.23372005210899548207540127094, 5.88629994554222809370700190006, 7.20245473816742579253515298607, 7.68432309366955493484463736552, 8.489326670914831272893639462866, 9.058059948754848271619358567625

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