Properties

Label 2-2268-63.16-c1-0-31
Degree $2$
Conductor $2268$
Sign $-0.902 - 0.430i$
Analytic cond. $18.1100$
Root an. cond. $4.25559$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 2.59i)7-s + (−2.5 − 4.33i)13-s + (−4 + 6.92i)19-s − 5·25-s + (3.5 − 6.06i)31-s + (−5.5 + 9.52i)37-s + (−2.5 + 4.33i)43-s + (−6.5 + 2.59i)49-s + (6.5 + 11.2i)61-s + (−2.5 + 4.33i)67-s + (5 + 8.66i)73-s + (−8.5 − 14.7i)79-s + (−10.0 + 8.66i)91-s + (−2.5 + 4.33i)97-s − 13·103-s + ⋯
L(s)  = 1  + (−0.188 − 0.981i)7-s + (−0.693 − 1.20i)13-s + (−0.917 + 1.58i)19-s − 25-s + (0.628 − 1.08i)31-s + (−0.904 + 1.56i)37-s + (−0.381 + 0.660i)43-s + (−0.928 + 0.371i)49-s + (0.832 + 1.44i)61-s + (−0.305 + 0.529i)67-s + (0.585 + 1.01i)73-s + (−0.956 − 1.65i)79-s + (−1.04 + 0.907i)91-s + (−0.253 + 0.439i)97-s − 1.28·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.902 - 0.430i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.902 - 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.902 - 0.430i$
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: \(1\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ -0.902 - 0.430i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 2.59i)T \)
good5 \( 1 + 5T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + (2.5 + 4.33i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4 - 6.92i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.5 + 6.06i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.5 - 9.52i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.5 - 4.33i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.5 - 11.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.5 - 4.33i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-5 - 8.66i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.5 + 14.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.5 - 4.33i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.248180674756241043012635287413, −7.974219144790363857080485495159, −7.09033070857923729957216668494, −6.22774701438591327642816962858, −5.48037511013558107636553176438, −4.41994130032382546527428593129, −3.71590111278700895791069820536, −2.71004839031671220607260072264, −1.41366516951239126085204772457, 0, 1.93043184035947133892744328242, 2.58917857763363884908221171203, 3.80325390648449771627720202081, 4.77011460718474810972073520259, 5.43653304431771675158668853595, 6.54365898447451057239060702590, 6.92009248738926261718539895868, 8.011480584360809180403525386329, 8.933098040709245341301682144560

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