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Properties

Label 2-2268-21.17-c1-0-10
Degree $2$
Conductor $2268$
Sign $0.184 - 0.982i$
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  + (1.95 + 3.39i)5-s + (1.96 − 1.77i)7-s + (−3.19 − 1.84i)11-s + 0.554i·13-s + (−2.91 + 5.05i)17-s + (4.62 − 2.66i)19-s + (−1.96 + 1.13i)23-s + (−5.16 + 8.94i)25-s + 4.08i·29-s + (7.00 + 4.04i)31-s + (9.85 + 3.18i)35-s + (3.89 + 6.75i)37-s + 7.18·41-s + 1.50·43-s + (−1.41 − 2.44i)47-s + ⋯
L(s)  = 1  + (0.875 + 1.51i)5-s + (0.742 − 0.670i)7-s + (−0.964 − 0.556i)11-s + 0.153i·13-s + (−0.707 + 1.22i)17-s + (1.06 − 0.612i)19-s + (−0.410 + 0.237i)23-s + (−1.03 + 1.78i)25-s + 0.758i·29-s + (1.25 + 0.726i)31-s + (1.66 + 0.538i)35-s + (0.640 + 1.11i)37-s + 1.12·41-s + 0.229·43-s + (−0.206 − 0.357i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.184 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.184 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.184 - 0.982i$
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: \((2,\ 2268,\ (\ :1/2),\ 0.184 - 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.057534574\)
\(L(\frac12)\) \(\approx\) \(2.057534574\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.96 + 1.77i)T \)
good5 \( 1 + (-1.95 - 3.39i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (3.19 + 1.84i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 0.554iT - 13T^{2} \)
17 \( 1 + (2.91 - 5.05i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.62 + 2.66i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.96 - 1.13i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.08iT - 29T^{2} \)
31 \( 1 + (-7.00 - 4.04i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.89 - 6.75i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 7.18T + 41T^{2} \)
43 \( 1 - 1.50T + 43T^{2} \)
47 \( 1 + (1.41 + 2.44i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.0415 - 0.0239i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.45 - 7.71i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.03 - 3.48i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.587 - 1.01i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.71iT - 71T^{2} \)
73 \( 1 + (3.52 + 2.03i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.97 - 3.41i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 7.69T + 83T^{2} \)
89 \( 1 + (-2.71 - 4.69i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 16.1iT - 97T^{2} \)
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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

−9.285087915593589550403037442702, −8.278593424416294182989064104693, −7.58978421200468231823927227691, −6.84151980454907039162871516797, −6.15752885111778146848827678261, −5.36251996975215882016440103978, −4.35764201168371710413988885289, −3.20728295048491925769587570573, −2.53292408008482903923241476425, −1.38849509751885392743230300465, 0.72487938883982599095852634698, 1.96774583868288919523859251657, 2.63343857762014666980119445823, 4.40195836534486355716538514830, 4.89568784756591658945164543019, 5.56014428647582695017120270689, 6.21508823507124305898397402422, 7.77492835567112890928158602891, 7.915173098344949232815293304696, 9.059460938598518398862331802446

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