Properties

Label 2-2268-21.17-c1-0-29
Degree $2$
Conductor $2268$
Sign $-0.990 + 0.135i$
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  + (−0.842 − 1.45i)5-s + (2.30 + 1.30i)7-s + (−3.38 − 1.95i)11-s − 6.05i·13-s + (−0.201 + 0.348i)17-s + (−0.145 + 0.0840i)19-s + (−7.69 + 4.44i)23-s + (1.07 − 1.86i)25-s + 7.10i·29-s + (−5.44 − 3.14i)31-s + (−0.0398 − 4.45i)35-s + (3.13 + 5.42i)37-s − 3.29·41-s − 3.60·43-s + (−4.38 − 7.59i)47-s + ⋯
L(s)  = 1  + (−0.376 − 0.652i)5-s + (0.870 + 0.492i)7-s + (−1.01 − 0.588i)11-s − 1.67i·13-s + (−0.0488 + 0.0845i)17-s + (−0.0334 + 0.0192i)19-s + (−1.60 + 0.926i)23-s + (0.215 − 0.373i)25-s + 1.31i·29-s + (−0.977 − 0.564i)31-s + (−0.00673 − 0.753i)35-s + (0.514 + 0.891i)37-s − 0.514·41-s − 0.550·43-s + (−0.639 − 1.10i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.135i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.135i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.990 + 0.135i$
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: \((2,\ 2268,\ (\ :1/2),\ -0.990 + 0.135i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4912400111\)
\(L(\frac12)\) \(\approx\) \(0.4912400111\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.30 - 1.30i)T \)
good5 \( 1 + (0.842 + 1.45i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (3.38 + 1.95i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 6.05iT - 13T^{2} \)
17 \( 1 + (0.201 - 0.348i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.145 - 0.0840i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (7.69 - 4.44i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.10iT - 29T^{2} \)
31 \( 1 + (5.44 + 3.14i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.13 - 5.42i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 3.29T + 41T^{2} \)
43 \( 1 + 3.60T + 43T^{2} \)
47 \( 1 + (4.38 + 7.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.94 - 2.85i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.25 - 3.89i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.43 - 2.56i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.95 + 5.11i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 11.4iT - 71T^{2} \)
73 \( 1 + (6.05 + 3.49i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.603 + 1.04i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 0.362T + 83T^{2} \)
89 \( 1 + (1.38 + 2.39i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 0.587iT - 97T^{2} \)
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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

−8.301501122675505940775781685980, −8.146940779861641876086522836605, −7.43329018136293206922119363763, −6.02403983100956513332583586886, −5.38885408238027551755078710755, −4.87240380981019849455935874342, −3.68880774058431601013667691937, −2.76892131441237639643734427257, −1.55199985106914609339489991173, −0.16028245909201567507987976487, 1.75114776372090553859877468036, 2.52252130156810448666679432007, 3.93157130481422421640835373004, 4.41430048100275552012322936219, 5.35923350492801453524759641205, 6.47160978873321350154991472318, 7.14514889866415689838727609745, 7.79266780700399526979602998025, 8.443154983307577018357475794339, 9.473865714467932496805532087170

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