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Properties

Label 2-1323-63.5-c1-0-35
Degree $2$
Conductor $1323$
Sign $0.0221 - 0.999i$
Analytic cond. $10.5642$
Root an. cond. $3.25026$
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  − 1.86i·2-s − 1.49·4-s + (−1.25 − 2.17i)5-s − 0.947i·8-s + (−4.05 + 2.34i)10-s + (−4.85 − 2.80i)11-s + (0.384 + 0.221i)13-s − 4.75·16-s + (1.53 + 2.66i)17-s + (−2.22 − 1.28i)19-s + (1.87 + 3.23i)20-s + (−5.24 + 9.07i)22-s + (6.83 − 3.94i)23-s + (−0.639 + 1.10i)25-s + (0.414 − 0.718i)26-s + ⋯
L(s)  = 1  − 1.32i·2-s − 0.746·4-s + (−0.560 − 0.970i)5-s − 0.335i·8-s + (−1.28 + 0.740i)10-s + (−1.46 − 0.845i)11-s + (0.106 + 0.0615i)13-s − 1.18·16-s + (0.373 + 0.646i)17-s + (−0.511 − 0.295i)19-s + (0.418 + 0.724i)20-s + (−1.11 + 1.93i)22-s + (1.42 − 0.822i)23-s + (−0.127 + 0.221i)25-s + (0.0813 − 0.140i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0221 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0221 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.0221 - 0.999i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (1097, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ 0.0221 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6183502671\)
\(L(\frac12)\) \(\approx\) \(0.6183502671\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + 1.86iT - 2T^{2} \)
5 \( 1 + (1.25 + 2.17i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (4.85 + 2.80i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.384 - 0.221i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.53 - 2.66i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.22 + 1.28i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6.83 + 3.94i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.71 - 1.56i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 10.4iT - 31T^{2} \)
37 \( 1 + (-0.708 + 1.22i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.64 - 2.85i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.75 + 8.23i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 2.14T + 47T^{2} \)
53 \( 1 + (4.20 - 2.42i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 7.30T + 59T^{2} \)
61 \( 1 - 8.55iT - 61T^{2} \)
67 \( 1 + 1.86T + 67T^{2} \)
71 \( 1 + 2.95iT - 71T^{2} \)
73 \( 1 + (-7.37 + 4.25i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 0.574T + 79T^{2} \)
83 \( 1 + (4.23 + 7.33i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.78 - 6.56i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.22 - 1.86i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.827900191559408947776495792411, −8.679253573025089347189043257065, −7.61508087703951546521569674030, −6.53317684676628559862892892748, −5.24689623912066681592239819149, −4.62325589522953803275362036440, −3.49175623370187826934336884281, −2.76176949529277265694605657574, −1.41087570946585926969579462815, −0.25425222407341064836007027098, 2.30066896550329644170827885628, 3.28396876656760993037610497650, 4.64513153858744752052270929469, 5.35853501348481118971832039823, 6.29193603719724032956314719743, 7.15354187109999638489941660611, 7.62847336441525400965207907166, 8.119597557317689976662784736817, 9.335218175942284085961911701611, 10.09656315198645017194365012705

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