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Properties

Label 2-1323-63.5-c1-0-11
Degree $2$
Conductor $1323$
Sign $-0.603 + 0.797i$
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  − 2.59i·2-s − 4.72·4-s + (0.626 + 1.08i)5-s + 7.07i·8-s + (2.81 − 1.62i)10-s + (0.534 + 0.308i)11-s + (−1.06 − 0.613i)13-s + 8.88·16-s + (2.21 + 3.83i)17-s + (1.64 + 0.950i)19-s + (−2.96 − 5.12i)20-s + (0.799 − 1.38i)22-s + (4.11 − 2.37i)23-s + (1.71 − 2.97i)25-s + (−1.59 + 2.75i)26-s + ⋯
L(s)  = 1  − 1.83i·2-s − 2.36·4-s + (0.280 + 0.485i)5-s + 2.50i·8-s + (0.889 − 0.513i)10-s + (0.161 + 0.0929i)11-s + (−0.294 − 0.170i)13-s + 2.22·16-s + (0.537 + 0.930i)17-s + (0.377 + 0.218i)19-s + (−0.662 − 1.14i)20-s + (0.170 − 0.295i)22-s + (0.857 − 0.495i)23-s + (0.343 − 0.594i)25-s + (−0.312 + 0.540i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.603 + 0.797i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.603 + 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.603 + 0.797i$
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: \((2,\ 1323,\ (\ :1/2),\ -0.603 + 0.797i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.491257532\)
\(L(\frac12)\) \(\approx\) \(1.491257532\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.59iT - 2T^{2} \)
5 \( 1 + (-0.626 - 1.08i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.534 - 0.308i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.06 + 0.613i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.21 - 3.83i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.64 - 0.950i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.11 + 2.37i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5.07 + 2.93i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 2.48iT - 31T^{2} \)
37 \( 1 + (-1.33 + 2.30i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.09 - 3.63i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.24 + 3.89i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 7.61T + 47T^{2} \)
53 \( 1 + (-2.67 + 1.54i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 3.56T + 59T^{2} \)
61 \( 1 + 14.4iT - 61T^{2} \)
67 \( 1 - 13.6T + 67T^{2} \)
71 \( 1 - 10.4iT - 71T^{2} \)
73 \( 1 + (9.95 - 5.74i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 4.03T + 79T^{2} \)
83 \( 1 + (4.36 + 7.56i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.811 + 1.40i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.76 - 5.06i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.849954091935490907513973893520, −8.788188775544858626059964899681, −8.128069957861571683898613326544, −6.87480803379664843064525367503, −5.75477190630813755346000501525, −4.74290124454845163120012283956, −3.84800791979459450006777785378, −2.93698307366342966672626049748, −2.12955550114839927918230618816, −0.869709812626308996977382855745, 1.00041207897894746446512610112, 3.11186830351309837472216871245, 4.43876641736009640394080724339, 5.17694150658790011114115672562, 5.69689243218839525399851735257, 6.88006205084846876367922012837, 7.21549316405302960344261602257, 8.192174676965592342524651120307, 8.988587285811681922750093426291, 9.401414835885567536447683986705

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