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Properties

Label 2-2268-7.2-c1-0-28
Degree $2$
Conductor $2268$
Sign $-0.991 - 0.126i$
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  + (−1 + 1.73i)5-s + (0.5 − 2.59i)7-s + (−2 − 3.46i)11-s + 3·13-s + (−3.5 − 6.06i)17-s + (−2.5 + 4.33i)19-s + (−2 + 3.46i)23-s + (0.500 + 0.866i)25-s − 29-s + (1.5 + 2.59i)31-s + (4 + 3.46i)35-s + (−5.5 + 9.52i)37-s − 9·41-s + 5·43-s + (−1.5 + 2.59i)47-s + ⋯
L(s)  = 1  + (−0.447 + 0.774i)5-s + (0.188 − 0.981i)7-s + (−0.603 − 1.04i)11-s + 0.832·13-s + (−0.848 − 1.47i)17-s + (−0.573 + 0.993i)19-s + (−0.417 + 0.722i)23-s + (0.100 + 0.173i)25-s − 0.185·29-s + (0.269 + 0.466i)31-s + (0.676 + 0.585i)35-s + (−0.904 + 1.56i)37-s − 1.40·41-s + 0.762·43-s + (−0.218 + 0.378i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.991 - 0.126i$
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: \(1\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ -0.991 - 0.126i)\)

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 \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2 + 3.46i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 3T + 13T^{2} \)
17 \( 1 + (3.5 + 6.06i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.5 - 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + T + 29T^{2} \)
31 \( 1 + (-1.5 - 2.59i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.5 - 9.52i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 9T + 41T^{2} \)
43 \( 1 - 5T + 43T^{2} \)
47 \( 1 + (1.5 - 2.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.5 + 2.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.5 - 6.06i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.5 - 2.59i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.5 + 11.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + (3.5 + 6.06i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.5 + 7.79i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - T + 83T^{2} \)
89 \( 1 + (7.5 - 12.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 17T + 97T^{2} \)
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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

−8.429330843397724415384928552654, −7.87884874230545552450532946852, −7.04302612356111764154705568448, −6.46752750286264323400924120464, −5.46984948140791200345804814874, −4.49503671676152370154693441856, −3.52745746022227903370012181950, −2.99009812616566836319546321014, −1.46605579995747644585466527201, 0, 1.74424765817236204957901896480, 2.52569428731877918263221608383, 3.98730641678001008285252673794, 4.56050873962319769681726636428, 5.43248323373330458389999920710, 6.26142673981552441814272794276, 7.09640288877772191509778548224, 8.236501128480494015092734713852, 8.543934540086807277270237145478

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