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Properties

Label 2-2268-21.17-c1-0-17
Degree $2$
Conductor $2268$
Sign $0.624 + 0.780i$
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  + (1.00 + 1.73i)5-s + (−2.64 + 0.0655i)7-s + (−4.15 − 2.40i)11-s + 0.998i·13-s + (−0.0445 + 0.0772i)17-s + (3.68 − 2.12i)19-s + (0.839 − 0.484i)23-s + (0.492 − 0.853i)25-s − 3.38i·29-s + (−4.35 − 2.51i)31-s + (−2.76 − 4.52i)35-s + (−0.0675 − 0.117i)37-s + 11.2·41-s + 7.33·43-s + (−1.76 − 3.06i)47-s + ⋯
L(s)  = 1  + (0.447 + 0.775i)5-s + (−0.999 + 0.0247i)7-s + (−1.25 − 0.723i)11-s + 0.276i·13-s + (−0.0108 + 0.0187i)17-s + (0.846 − 0.488i)19-s + (0.175 − 0.101i)23-s + (0.0985 − 0.170i)25-s − 0.628i·29-s + (−0.782 − 0.451i)31-s + (−0.467 − 0.764i)35-s + (−0.0111 − 0.0192i)37-s + 1.75·41-s + 1.11·43-s + (−0.257 − 0.446i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 + 0.780i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.624 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.624 + 0.780i$
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: \((2,\ 2268,\ (\ :1/2),\ 0.624 + 0.780i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.254811975\)
\(L(\frac12)\) \(\approx\) \(1.254811975\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.64 - 0.0655i)T \)
good5 \( 1 + (-1.00 - 1.73i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (4.15 + 2.40i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 0.998iT - 13T^{2} \)
17 \( 1 + (0.0445 - 0.0772i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.68 + 2.12i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.839 + 0.484i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.38iT - 29T^{2} \)
31 \( 1 + (4.35 + 2.51i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.0675 + 0.117i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 11.2T + 41T^{2} \)
43 \( 1 - 7.33T + 43T^{2} \)
47 \( 1 + (1.76 + 3.06i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.31 + 3.07i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.88 - 8.45i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-11.2 + 6.51i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.57 + 13.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.56iT - 71T^{2} \)
73 \( 1 + (-3.73 - 2.15i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.94 + 10.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.24T + 83T^{2} \)
89 \( 1 + (-1.06 - 1.84i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 1.19iT - 97T^{2} \)
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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

−9.154250466157911333873990563422, −8.005592739075685126605001118342, −7.36104886125963797183440629101, −6.48116916924594151048231432289, −5.90438589640019912316469814264, −5.09894396632061051605241615670, −3.83587283260676795257419867941, −2.92192687309679478321587281868, −2.36561134037181895589751764913, −0.48944227890283956048366607269, 1.05935953902699681608312251231, 2.38830031415149704749361607745, 3.26794807982909674286897872963, 4.37551311800115576340785606774, 5.39540522624693372116454058309, 5.69833877104133839383623142435, 6.93145475573878772870001385212, 7.53686097749041780286575977535, 8.397514049370780895537731815907, 9.375526526206445483875064812383

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