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Properties

Label 2-2268-63.16-c1-0-3
Degree $2$
Conductor $2268$
Sign $-0.304 - 0.952i$
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  + 2.30·5-s + (−2.14 − 1.55i)7-s − 4.46·11-s + (1.42 + 2.46i)13-s + (0.115 + 0.199i)17-s + (−1.49 + 2.58i)19-s + 0.800·23-s + 0.299·25-s + (−3.82 + 6.62i)29-s + (−2.64 + 4.57i)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 + (2.88 + 5.00i)47-s + ⋯
L(s)  = 1  + 1.02·5-s + (−0.810 − 0.586i)7-s − 1.34·11-s + (0.394 + 0.682i)13-s + (0.0279 + 0.0484i)17-s + (−0.342 + 0.593i)19-s + 0.166·23-s + 0.0598·25-s + (−0.710 + 1.23i)29-s + (−0.474 + 0.822i)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 + (0.421 + 0.729i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.304 - 0.952i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.304 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.304 - 0.952i$
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: \((2,\ 2268,\ (\ :1/2),\ -0.304 - 0.952i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.015908511\)
\(L(\frac12)\) \(\approx\) \(1.015908511\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.14 + 1.55i)T \)
good5 \( 1 - 2.30T + 5T^{2} \)
11 \( 1 + 4.46T + 11T^{2} \)
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 \( 1 - 0.800T + 23T^{2} \)
29 \( 1 + (3.82 - 6.62i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.64 - 4.57i)T + (-15.5 - 26.8i)T^{2} \)
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 \( 1 + (-2.88 - 5.00i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.31 - 7.48i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.17 - 7.23i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.58 - 11.4i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.76 + 6.52i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.59T + 71T^{2} \)
73 \( 1 + (2.29 + 3.97i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.83 + 8.37i)T + (-39.5 + 68.4i)T^{2} \)
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

−9.204199519890100227814306981616, −8.708844954303846439390036612094, −7.51932334497364516420507375699, −6.98967954081557251488189283775, −5.98007362765474782989742272522, −5.54220606791809671310656311532, −4.43747221423680974612517948345, −3.43346747452777831602649641680, −2.48899143590471316417539637277, −1.41440040089737748040723810057, 0.32720666367031902302941417175, 2.13461242841696202437570008036, 2.68251391581053308881404487174, 3.76719300996904302336172582534, 5.07724810899710968193355086341, 5.73594167196842887659476621934, 6.18789521350819214962946102764, 7.25017466024892176233456118509, 8.069864130744940226787308312838, 8.882569578306863297468047356475

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