Properties

Label 2-1323-63.5-c1-0-23
Degree $2$
Conductor $1323$
Sign $0.385 + 0.922i$
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  − 0.718i·2-s + 1.48·4-s + (−0.723 − 1.25i)5-s − 2.50i·8-s + (−0.900 + 0.519i)10-s + (1.55 + 0.900i)11-s + (1.88 + 1.09i)13-s + 1.17·16-s + (1.95 + 3.38i)17-s + (3.47 + 2.00i)19-s + (−1.07 − 1.86i)20-s + (0.646 − 1.11i)22-s + (4.91 − 2.83i)23-s + (1.45 − 2.51i)25-s + (0.783 − 1.35i)26-s + ⋯
L(s)  = 1  − 0.507i·2-s + 0.742·4-s + (−0.323 − 0.560i)5-s − 0.884i·8-s + (−0.284 + 0.164i)10-s + (0.470 + 0.271i)11-s + (0.523 + 0.302i)13-s + 0.292·16-s + (0.473 + 0.820i)17-s + (0.797 + 0.460i)19-s + (−0.240 − 0.416i)20-s + (0.137 − 0.238i)22-s + (1.02 − 0.591i)23-s + (0.290 − 0.503i)25-s + (0.153 − 0.266i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.385 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.385 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.385 + 0.922i$
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: \((2,\ 1323,\ (\ :1/2),\ 0.385 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.121882997\)
\(L(\frac12)\) \(\approx\) \(2.121882997\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.718iT - 2T^{2} \)
5 \( 1 + (0.723 + 1.25i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.55 - 0.900i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.88 - 1.09i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.95 - 3.38i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.47 - 2.00i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.91 + 2.83i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (8.49 - 4.90i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 2.83iT - 31T^{2} \)
37 \( 1 + (0.411 - 0.713i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.90 + 10.2i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.76 + 6.52i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 2.33T + 47T^{2} \)
53 \( 1 + (0.996 - 0.575i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 9.79T + 59T^{2} \)
61 \( 1 + 2.35iT - 61T^{2} \)
67 \( 1 + 0.312T + 67T^{2} \)
71 \( 1 + 1.94iT - 71T^{2} \)
73 \( 1 + (2.42 - 1.40i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 12.4T + 79T^{2} \)
83 \( 1 + (-3.60 - 6.25i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (5.28 - 9.16i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-13.4 + 7.75i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.456504534000466574757323944832, −8.827078379496185214040760183100, −7.79965288074576225961654116152, −7.07912580516540208765639782985, −6.19856623190921702307453836977, −5.28164238575217607591378555496, −4.03481403171361955214859292124, −3.39297390974000354478023065151, −2.03522424496917841456392110580, −1.04124631085986240824948660111, 1.31090390575830888237564222385, 2.85589562567440483242476419786, 3.44046283462045309283037939708, 4.92203587189337933459865566679, 5.79341207836424429866462244047, 6.56117521888038655631513160299, 7.44415706457805933322780107657, 7.74651873704883084047739433334, 8.965840789887500112373087208151, 9.663394014650515044732023878260

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