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Properties

Label 2-2268-63.25-c1-0-9
Degree $2$
Conductor $2268$
Sign $-0.378 - 0.925i$
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  + (−1.15 + 1.99i)5-s + (2.41 + 1.08i)7-s + (2.23 + 3.86i)11-s + (1.42 + 2.46i)13-s + (0.115 − 0.199i)17-s + (−1.49 − 2.58i)19-s + (−0.400 + 0.693i)23-s + (−0.149 − 0.259i)25-s + (−3.82 + 6.62i)29-s + 5.28·31-s + (−4.93 + 3.57i)35-s + (−1.69 − 2.93i)37-s + (0.899 + 1.55i)41-s + (4.85 − 8.41i)43-s − 5.77·47-s + ⋯
L(s)  = 1  + (−0.514 + 0.891i)5-s + (0.912 + 0.408i)7-s + (0.672 + 1.16i)11-s + (0.394 + 0.682i)13-s + (0.0279 − 0.0484i)17-s + (−0.342 − 0.593i)19-s + (−0.0834 + 0.144i)23-s + (−0.0299 − 0.0518i)25-s + (−0.710 + 1.23i)29-s + 0.949·31-s + (−0.833 + 0.603i)35-s + (−0.278 − 0.482i)37-s + (0.140 + 0.243i)41-s + (0.740 − 1.28i)43-s − 0.842·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.378 - 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.378 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.378 - 0.925i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ -0.378 - 0.925i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.689418572\)
\(L(\frac12)\) \(\approx\) \(1.689418572\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.41 - 1.08i)T \)
good5 \( 1 + (1.15 - 1.99i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.23 - 3.86i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.42 - 2.46i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.115 + 0.199i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.49 + 2.58i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.400 - 0.693i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.82 - 6.62i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.28T + 31T^{2} \)
37 \( 1 + (1.69 + 2.93i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.899 - 1.55i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.85 + 8.41i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 5.77T + 47T^{2} \)
53 \( 1 + (-4.31 + 7.48i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 8.35T + 59T^{2} \)
61 \( 1 + 13.1T + 61T^{2} \)
67 \( 1 + 7.53T + 67T^{2} \)
71 \( 1 + 8.59T + 71T^{2} \)
73 \( 1 + (2.29 - 3.97i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 9.67T + 79T^{2} \)
83 \( 1 + (8.46 - 14.6i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.944 + 1.63i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.70 + 13.3i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.098917049911228904832142557043, −8.607079476143622861600783116940, −7.48981122296945629378389769528, −7.09332584457124693538199037641, −6.31803485119682266621151371647, −5.19451434515048130318815915064, −4.40059745008179497509838447813, −3.61259980434922460952737240315, −2.42861687939592033501986130015, −1.52843663236075888137698283441, 0.62606426718272936348018197809, 1.51021648838913296509927552052, 3.04291848444461674386414371338, 4.09843790671086980179289589112, 4.56928322385054754262028552767, 5.68885005358535574077635251808, 6.25120460207835987739693555617, 7.51948171495025074107682368640, 8.189657384317868724403133365818, 8.515705828718837711385299280620

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