Properties

Label 2-2268-63.16-c1-0-4
Degree $2$
Conductor $2268$
Sign $0.997 + 0.0746i$
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  − 4.28·5-s + (−2.64 + 0.0963i)7-s − 3.81·11-s + (−1.64 − 2.84i)13-s + (0.405 + 0.702i)17-s + (−3.54 + 6.14i)19-s − 6.47·23-s + 13.3·25-s + (−1.90 + 3.30i)29-s + (−1.64 + 2.84i)31-s + (11.3 − 0.413i)35-s + (2.88 − 4.99i)37-s + (−1.04 − 1.81i)41-s + (4.38 − 7.59i)43-s + (−1.66 − 2.88i)47-s + ⋯
L(s)  = 1  − 1.91·5-s + (−0.999 + 0.0364i)7-s − 1.14·11-s + (−0.455 − 0.789i)13-s + (0.0983 + 0.170i)17-s + (−0.814 + 1.41i)19-s − 1.35·23-s + 2.67·25-s + (−0.353 + 0.612i)29-s + (−0.295 + 0.511i)31-s + (1.91 − 0.0698i)35-s + (0.473 − 0.820i)37-s + (−0.163 − 0.283i)41-s + (0.668 − 1.15i)43-s + (−0.243 − 0.421i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0746i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.997 + 0.0746i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ 0.997 + 0.0746i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3491059501\)
\(L(\frac12)\) \(\approx\) \(0.3491059501\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.64 - 0.0963i)T \)
good5 \( 1 + 4.28T + 5T^{2} \)
11 \( 1 + 3.81T + 11T^{2} \)
13 \( 1 + (1.64 + 2.84i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.405 - 0.702i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.54 - 6.14i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 6.47T + 23T^{2} \)
29 \( 1 + (1.90 - 3.30i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.64 - 2.84i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.88 + 4.99i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.04 + 1.81i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.38 + 7.59i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.66 + 2.88i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.93 + 8.54i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.73 - 3.01i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.97 + 5.15i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.76 + 3.05i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.05T + 71T^{2} \)
73 \( 1 + (-5.19 - 8.99i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.57 - 4.45i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.856 + 1.48i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (6.26 - 10.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.522 - 0.905i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.775425587523751051141607460037, −8.037534237990451337846305949215, −7.68900619681267526585478405689, −6.87007757702330163201988772167, −5.85134423092258886929202736447, −4.99112234572386721359957159705, −3.79776978188364111627291731080, −3.55107608500768536322336209543, −2.37124703525092046313598063037, −0.34376344571732826206892215193, 0.36899925280149575790820288373, 2.48952653441471059796971726951, 3.24955355096492717324696393651, 4.29887712250470843265532209905, 4.66177332590160705909573281388, 6.05430101450892640923955792376, 6.85236967142430341400187875726, 7.62122615495382770219276451858, 8.037079514111233713699028420204, 8.991182824252928746564568790002

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