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Properties

Label 2-2268-7.2-c1-0-9
Degree $2$
Conductor $2268$
Sign $0.256 - 0.966i$
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  + (−1.26 + 2.18i)5-s + (−2.50 − 0.839i)7-s + (−0.687 − 1.18i)11-s + 5.60·13-s + (2.69 + 4.66i)17-s + (2.44 − 4.23i)19-s + (2.08 − 3.61i)23-s + (−0.675 − 1.17i)25-s − 3.13·29-s + (−2.40 − 4.15i)31-s + (4.99 − 4.41i)35-s + (−2.69 + 4.67i)37-s − 6.05·41-s − 4.89·43-s + (−2.82 + 4.89i)47-s + ⋯
L(s)  = 1  + (−0.563 + 0.976i)5-s + (−0.948 − 0.317i)7-s + (−0.207 − 0.358i)11-s + 1.55·13-s + (0.653 + 1.13i)17-s + (0.561 − 0.972i)19-s + (0.435 − 0.753i)23-s + (−0.135 − 0.234i)25-s − 0.582·29-s + (−0.431 − 0.746i)31-s + (0.844 − 0.746i)35-s + (−0.443 + 0.768i)37-s − 0.946·41-s − 0.746·43-s + (−0.412 + 0.714i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.256 - 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.256 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.256 - 0.966i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (1297, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ 0.256 - 0.966i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.285836708\)
\(L(\frac12)\) \(\approx\) \(1.285836708\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.50 + 0.839i)T \)
good5 \( 1 + (1.26 - 2.18i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.687 + 1.18i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 5.60T + 13T^{2} \)
17 \( 1 + (-2.69 - 4.66i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.44 + 4.23i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.08 + 3.61i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.13T + 29T^{2} \)
31 \( 1 + (2.40 + 4.15i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.69 - 4.67i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 6.05T + 41T^{2} \)
43 \( 1 + 4.89T + 43T^{2} \)
47 \( 1 + (2.82 - 4.89i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-7.00 - 12.1i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-7.13 - 12.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.42 + 5.93i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.05 - 7.02i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.25T + 71T^{2} \)
73 \( 1 + (-3.51 - 6.08i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.37 - 2.38i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 14.9T + 83T^{2} \)
89 \( 1 + (2.75 - 4.77i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 1.78T + 97T^{2} \)
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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

−9.105196446610748041642636349765, −8.407658882831399845529218918663, −7.57017869794162991754641105229, −6.77156561875936155474838510880, −6.26936647209924396726357105517, −5.38384073476103397269604960140, −3.94824655856742945150844758142, −3.50068510891799140135240735771, −2.70603251877323211462286497805, −1.04331822762229550964097805499, 0.54949357346831254051921424704, 1.77143644891930864017450804028, 3.43367153473769938545217073249, 3.64281685934941251263934028736, 5.09879929356779432574184332669, 5.48789607976764296975825152739, 6.57529466241199894563182697719, 7.33767541824723733842934597645, 8.261109036627417527605795295037, 8.786297596344336386396801805339

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