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Properties

Label 2-2268-21.17-c1-0-3
Degree $2$
Conductor $2268$
Sign $0.315 - 0.948i$
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  + (−1.48 − 2.57i)5-s + (−2.18 + 1.49i)7-s + (4.09 + 2.36i)11-s − 4.08i·13-s + (−0.835 + 1.44i)17-s + (−4.25 + 2.45i)19-s + (−4.25 + 2.45i)23-s + (−1.91 + 3.30i)25-s − 0.275i·29-s + (1.38 + 0.801i)31-s + (7.08 + 3.40i)35-s + (−1.69 − 2.93i)37-s + 7.11·41-s − 10.4·43-s + (5.49 + 9.52i)47-s + ⋯
L(s)  = 1  + (−0.664 − 1.15i)5-s + (−0.825 + 0.564i)7-s + (1.23 + 0.712i)11-s − 1.13i·13-s + (−0.202 + 0.350i)17-s + (−0.975 + 0.563i)19-s + (−0.886 + 0.511i)23-s + (−0.382 + 0.661i)25-s − 0.0511i·29-s + (0.249 + 0.143i)31-s + (1.19 + 0.575i)35-s + (−0.278 − 0.483i)37-s + 1.11·41-s − 1.59·43-s + (0.802 + 1.38i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.315 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.315 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.315 - 0.948i$
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: \((2,\ 2268,\ (\ :1/2),\ 0.315 - 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8762256633\)
\(L(\frac12)\) \(\approx\) \(0.8762256633\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.18 - 1.49i)T \)
good5 \( 1 + (1.48 + 2.57i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-4.09 - 2.36i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 4.08iT - 13T^{2} \)
17 \( 1 + (0.835 - 1.44i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.25 - 2.45i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.25 - 2.45i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.275iT - 29T^{2} \)
31 \( 1 + (-1.38 - 0.801i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.69 + 2.93i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 7.11T + 41T^{2} \)
43 \( 1 + 10.4T + 43T^{2} \)
47 \( 1 + (-5.49 - 9.52i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.707 - 0.408i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.37 - 2.38i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.23 + 3.60i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.80 - 10.0i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.4iT - 71T^{2} \)
73 \( 1 + (-13.6 - 7.88i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.15 - 10.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 8.07T + 83T^{2} \)
89 \( 1 + (-4.60 - 7.98i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 8.09iT - 97T^{2} \)
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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

−9.133782055787798233146265521909, −8.414893077933915314575364665738, −7.85957996816750108791044095352, −6.77440520955830855802071679773, −6.03080862833854464530320713127, −5.23178375574838875865903117738, −4.16638791506610097211341621984, −3.71118721151742891399681893617, −2.33603021366660940710563189366, −1.07169189452388431478348008171, 0.34740718249722135407090889959, 2.07203908360582233476087489028, 3.24754487980293897243575095962, 3.83987898616740940586233413824, 4.56521522862857567614180594669, 6.17870381820314971027510652602, 6.62496685221450605689918244206, 7.03254517519833082084195980982, 8.057174571304897117444918510698, 8.957228412632442948969772383752

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