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Properties

Label 2-2268-63.4-c1-0-31
Degree $2$
Conductor $2268$
Sign $-0.975 - 0.220i$
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 + (0.5 + 0.866i)19-s − 5·25-s + (−5.5 − 9.52i)31-s + (−5.5 − 9.52i)37-s + (6.5 + 11.2i)43-s + (−6.5 + 2.59i)49-s + (−7 + 12.1i)61-s + (−2.5 − 4.33i)67-s + (−8.5 + 14.7i)73-s + (−8.5 + 14.7i)79-s + (12.5 + 4.33i)91-s + (−7 − 12.1i)97-s − 13·103-s + ⋯
L(s)  = 1  + (−0.188 − 0.981i)7-s + (−0.693 + 1.20i)13-s + (0.114 + 0.198i)19-s − 25-s + (−0.987 − 1.71i)31-s + (−0.904 − 1.56i)37-s + (0.991 + 1.71i)43-s + (−0.928 + 0.371i)49-s + (−0.896 + 1.55i)61-s + (−0.305 − 0.529i)67-s + (−0.994 + 1.72i)73-s + (−0.956 + 1.65i)79-s + (1.31 + 0.453i)91-s + (−0.710 − 1.23i)97-s − 1.28·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 - 0.220i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.975 - 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.975 - 0.220i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ -0.975 - 0.220i)\)

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 \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (5.5 + 9.52i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.5 + 9.52i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.5 - 11.2i)T + (-21.5 + 37.2i)T^{2} \)
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 \( 1 + (7 - 12.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.5 + 4.33i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (8.5 - 14.7i)T + (-36.5 - 63.2i)T^{2} \)
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 \( 1 + (7 + 12.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.671131994267711853551069541782, −7.47543495519043085394123869258, −7.36288626373026552746016215448, −6.28794468224460445429142665671, −5.50973610260415290305082869205, −4.25634642556941046501974163839, −3.99133648412656013929948771383, −2.62253147020274484765712305503, −1.54411190143584267816944787932, 0, 1.73243019184440802996069156400, 2.81474833742590553500052434650, 3.52926385911289743124619338202, 4.90347528199009248676168407541, 5.42300451138715627894427055189, 6.23606275897716581455959955764, 7.18168453819984626874993426681, 7.916986045840507899519835116874, 8.735337316128786531026605872727

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