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Properties

Label 2-2268-21.17-c1-0-15
Degree $2$
Conductor $2268$
Sign $0.624 + 0.780i$
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  + (−1.00 − 1.73i)5-s + (−2.64 + 0.0655i)7-s + (4.15 + 2.40i)11-s + 0.998i·13-s + (0.0445 − 0.0772i)17-s + (3.68 − 2.12i)19-s + (−0.839 + 0.484i)23-s + (0.492 − 0.853i)25-s + 3.38i·29-s + (−4.35 − 2.51i)31-s + (2.76 + 4.52i)35-s + (−0.0675 − 0.117i)37-s − 11.2·41-s + 7.33·43-s + (1.76 + 3.06i)47-s + ⋯
L(s)  = 1  + (−0.447 − 0.775i)5-s + (−0.999 + 0.0247i)7-s + (1.25 + 0.723i)11-s + 0.276i·13-s + (0.0108 − 0.0187i)17-s + (0.846 − 0.488i)19-s + (−0.175 + 0.101i)23-s + (0.0985 − 0.170i)25-s + 0.628i·29-s + (−0.782 − 0.451i)31-s + (0.467 + 0.764i)35-s + (−0.0111 − 0.0192i)37-s − 1.75·41-s + 1.11·43-s + (0.257 + 0.446i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 + 0.780i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.624 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.624 + 0.780i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (1781, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ 0.624 + 0.780i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.408812770\)
\(L(\frac12)\) \(\approx\) \(1.408812770\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.64 - 0.0655i)T \)
good5 \( 1 + (1.00 + 1.73i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-4.15 - 2.40i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 0.998iT - 13T^{2} \)
17 \( 1 + (-0.0445 + 0.0772i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.68 + 2.12i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.839 - 0.484i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 3.38iT - 29T^{2} \)
31 \( 1 + (4.35 + 2.51i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.0675 + 0.117i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 11.2T + 41T^{2} \)
43 \( 1 - 7.33T + 43T^{2} \)
47 \( 1 + (-1.76 - 3.06i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.31 - 3.07i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.88 + 8.45i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-11.2 + 6.51i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.57 + 13.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.56iT - 71T^{2} \)
73 \( 1 + (-3.73 - 2.15i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.94 + 10.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.24T + 83T^{2} \)
89 \( 1 + (1.06 + 1.84i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 1.19iT - 97T^{2} \)
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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

−9.071071749210324185026879157932, −8.264550309658191306709920212846, −7.16561775246882083767400147528, −6.77647157651036452415645510698, −5.76469275862083646164697294397, −4.82762595780422409817641529271, −4.01791832561760453220697112515, −3.27948935161328281803935849304, −1.90247432923441240009791610590, −0.63683166301827646020471717463, 0.960047812704914356800512619972, 2.54335292293225370212705110130, 3.63864856513276878745749738299, 3.75919946274638844155760458928, 5.35869433633836208115649483154, 6.10468583590347405759379470439, 6.88589860416238642623425061999, 7.35635899233359762942360021596, 8.471120523076945509270082915443, 9.088724020351578058412230996037

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