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Properties

Label 2-2268-63.41-c1-0-10
Degree $2$
Conductor $2268$
Sign $0.630 - 0.775i$
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  + (0.5 + 2.59i)7-s + (1.5 + 0.866i)13-s − 8.66i·19-s + (2.5 + 4.33i)25-s + (9 + 5.19i)31-s + 37-s + (4 + 6.92i)43-s + (−6.5 + 2.59i)49-s + (−7.5 + 4.33i)61-s + (−5.5 + 9.52i)67-s + 1.73i·73-s + (6.5 + 11.2i)79-s + (−1.5 + 4.33i)91-s + (16.5 − 9.52i)97-s + (16.5 + 9.52i)103-s + ⋯
L(s)  = 1  + (0.188 + 0.981i)7-s + (0.416 + 0.240i)13-s − 1.98i·19-s + (0.5 + 0.866i)25-s + (1.61 + 0.933i)31-s + 0.164·37-s + (0.609 + 1.05i)43-s + (−0.928 + 0.371i)49-s + (−0.960 + 0.554i)61-s + (−0.671 + 1.16i)67-s + 0.202i·73-s + (0.731 + 1.26i)79-s + (−0.157 + 0.453i)91-s + (1.67 − 0.967i)97-s + (1.62 + 0.938i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.630 - 0.775i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.630 - 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.630 - 0.775i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (377, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ 0.630 - 0.775i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.800930228\)
\(L(\frac12)\) \(\approx\) \(1.800930228\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.5 - 0.866i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 8.66iT - 19T^{2} \)
23 \( 1 + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-9 - 5.19i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - T + 37T^{2} \)
41 \( 1 + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4 - 6.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.5 - 4.33i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.5 - 9.52i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 1.73iT - 73T^{2} \)
79 \( 1 + (-6.5 - 11.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (-16.5 + 9.52i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.957845594041777441685976503606, −8.614880188907827938813873798234, −7.58884737148203376123949510851, −6.74191425909987842746220278727, −6.05344493236716187471932821006, −5.07074140562705290133021837929, −4.50837833703573721072753568677, −3.14281811639296592433677437549, −2.46893187669859520433949196678, −1.13158635465035578929384526109, 0.73437428858594037551216184151, 1.90042347101419496517922453147, 3.23142853138586706696554153105, 4.04414834191013381453118875332, 4.77873509028592972501557530030, 5.95103662025693935911652799337, 6.45475002832406033965570038753, 7.60180260246159955692598488342, 7.977926946483047116697374987659, 8.816461954399482816526542109352

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