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Properties

Label 2-2268-21.5-c1-0-31
Degree $2$
Conductor $2268$
Sign $-0.840 + 0.542i$
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  + (1.37 − 2.37i)5-s + (0.900 − 2.48i)7-s + (0.362 − 0.209i)11-s − 1.53i·13-s + (−1.95 − 3.38i)17-s + (−5.11 − 2.95i)19-s + (−7.72 − 4.46i)23-s + (−1.26 − 2.18i)25-s + 6.93i·29-s + (3.05 − 1.76i)31-s + (−4.67 − 5.55i)35-s + (−4.54 + 7.87i)37-s − 2.12·41-s + 11.5·43-s + (0.885 − 1.53i)47-s + ⋯
L(s)  = 1  + (0.613 − 1.06i)5-s + (0.340 − 0.940i)7-s + (0.109 − 0.0630i)11-s − 0.424i·13-s + (−0.473 − 0.820i)17-s + (−1.17 − 0.678i)19-s + (−1.61 − 0.930i)23-s + (−0.252 − 0.437i)25-s + 1.28i·29-s + (0.548 − 0.316i)31-s + (−0.790 − 0.938i)35-s + (−0.747 + 1.29i)37-s − 0.331·41-s + 1.76·43-s + (0.129 − 0.223i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.840 + 0.542i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.840 + 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.840 + 0.542i$
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: \((2,\ 2268,\ (\ :1/2),\ -0.840 + 0.542i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.451520340\)
\(L(\frac12)\) \(\approx\) \(1.451520340\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.900 + 2.48i)T \)
good5 \( 1 + (-1.37 + 2.37i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.362 + 0.209i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.53iT - 13T^{2} \)
17 \( 1 + (1.95 + 3.38i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.11 + 2.95i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (7.72 + 4.46i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 6.93iT - 29T^{2} \)
31 \( 1 + (-3.05 + 1.76i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.54 - 7.87i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.12T + 41T^{2} \)
43 \( 1 - 11.5T + 43T^{2} \)
47 \( 1 + (-0.885 + 1.53i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.39 - 1.96i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.02 - 3.51i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.61 - 0.932i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.38 - 11.0i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.51iT - 71T^{2} \)
73 \( 1 + (-1.65 + 0.952i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.433 + 0.751i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6.91T + 83T^{2} \)
89 \( 1 + (4.88 - 8.46i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 0.231iT - 97T^{2} \)
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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

−8.644518075879388682564367185361, −8.148234993675702564091241595015, −7.10229883820201416505340584350, −6.40866733303688813342743690412, −5.41761749977485489282664771277, −4.64702623411758992300383998563, −4.08807051801754043712122807680, −2.67572655282926689652570206384, −1.57152656510487430226112807327, −0.46702282375207239999040605727, 1.99478942674980627258902369928, 2.25380223705242902464822351855, 3.62262407406412572861268042396, 4.42620421499087252654931365643, 5.82611855023846803609740821345, 6.02456858504023976404582465191, 6.86316784858754830349293598280, 7.907680270721417663376087486670, 8.504780204923357969498235550514, 9.425911856687196788221176355886

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