Properties

Label 2-2268-63.4-c1-0-14
Degree $2$
Conductor $2268$
Sign $0.678 - 0.734i$
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  + 3·5-s + (2.5 + 0.866i)7-s − 3·11-s + (−1 + 1.73i)13-s + (−1.5 + 2.59i)17-s + (0.5 + 0.866i)19-s + 3·23-s + 4·25-s + (3 + 5.19i)29-s + (3.5 + 6.06i)31-s + (7.5 + 2.59i)35-s + (0.5 + 0.866i)37-s + (−3 + 5.19i)41-s + (2 + 3.46i)43-s + (4.5 − 7.79i)47-s + ⋯
L(s)  = 1  + 1.34·5-s + (0.944 + 0.327i)7-s − 0.904·11-s + (−0.277 + 0.480i)13-s + (−0.363 + 0.630i)17-s + (0.114 + 0.198i)19-s + 0.625·23-s + 0.800·25-s + (0.557 + 0.964i)29-s + (0.628 + 1.08i)31-s + (1.26 + 0.439i)35-s + (0.0821 + 0.142i)37-s + (−0.468 + 0.811i)41-s + (0.304 + 0.528i)43-s + (0.656 − 1.13i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.678 - 0.734i$
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: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ 0.678 - 0.734i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.394052064\)
\(L(\frac12)\) \(\approx\) \(2.394052064\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 - 0.866i)T \)
good5 \( 1 - 3T + 5T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 3T + 23T^{2} \)
29 \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.5 - 6.06i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3 - 5.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2 - 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.5 + 7.79i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.5 - 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.5 + 7.79i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.5 - 6.06i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.5 + 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6 + 10.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (7.5 + 12.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5 - 8.66i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.996709470845629240757928136072, −8.532629305506617940502958397242, −7.61894333420969501320239071835, −6.69287195930079397327095657453, −5.93354554291838312613028279328, −5.11177778832526010162408683924, −4.64623215368187315270055912545, −3.11110187879633146575881872589, −2.17897767403272087355911664138, −1.41619095905830985023074902380, 0.848039094006104399924121059848, 2.17438803483430949050960317334, 2.72661340318358794817875419747, 4.24882118858873434484502336128, 5.11077792079926952798338981491, 5.60583023765135065590844042035, 6.53553974454786450805556713423, 7.47780898658539831513244281411, 8.077640387632472704353197694227, 9.012611170970182480653206575889

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