Properties

Label 2-1323-63.25-c1-0-23
Degree $2$
Conductor $1323$
Sign $0.104 + 0.994i$
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  − 0.0683·2-s − 1.99·4-s + (1.33 − 2.30i)5-s + 0.273·8-s + (−0.0910 + 0.157i)10-s + (−0.799 − 1.38i)11-s + (2.62 + 4.54i)13-s + 3.97·16-s + (3.27 − 5.67i)17-s + (0.950 + 1.64i)19-s + (−2.65 + 4.60i)20-s + (0.0546 + 0.0946i)22-s + (−1.53 + 2.65i)23-s + (−1.04 − 1.81i)25-s + (−0.179 − 0.311i)26-s + ⋯
L(s)  = 1  − 0.0483·2-s − 0.997·4-s + (0.595 − 1.03i)5-s + 0.0965·8-s + (−0.0287 + 0.0498i)10-s + (−0.241 − 0.417i)11-s + (0.728 + 1.26i)13-s + 0.992·16-s + (0.793 − 1.37i)17-s + (0.218 + 0.377i)19-s + (−0.594 + 1.02i)20-s + (0.0116 + 0.0201i)22-s + (−0.319 + 0.554i)23-s + (−0.209 − 0.363i)25-s + (−0.0352 − 0.0610i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.104 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.104 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.104 + 0.994i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (802, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ 0.104 + 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.310017679\)
\(L(\frac12)\) \(\approx\) \(1.310017679\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.0683T + 2T^{2} \)
5 \( 1 + (-1.33 + 2.30i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.799 + 1.38i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.62 - 4.54i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.27 + 5.67i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.950 - 1.64i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.53 - 2.65i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.19 + 5.53i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.71T + 31T^{2} \)
37 \( 1 + (2.11 + 3.66i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.69 + 6.40i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.63 + 9.75i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 3.79T + 47T^{2} \)
53 \( 1 + (-4.44 + 7.70i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 10.8T + 59T^{2} \)
61 \( 1 + 2.71T + 61T^{2} \)
67 \( 1 + 3.32T + 67T^{2} \)
71 \( 1 - 12.3T + 71T^{2} \)
73 \( 1 + (-1.09 + 1.90i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 0.813T + 79T^{2} \)
83 \( 1 + (3.41 - 5.92i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.235 + 0.407i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.57 - 4.46i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.265222711204286734699720594771, −8.905041740551434012388261772452, −8.035653542939511164995125447452, −7.07707299814562353166052175222, −5.69083218957057895902512661373, −5.37720140180775715984530317192, −4.34476892406061073442857384322, −3.51384841310972012145443398766, −1.84877102031256323329714797264, −0.64716259424918804965618352357, 1.30672010685751283806872801435, 2.87168895159896928500556921410, 3.61794322766920731355332085289, 4.79109780840530959981665211952, 5.74760379030360938551032695523, 6.33387451592552979474463060650, 7.53888394357503980544341023582, 8.203574555022019371292763433584, 9.030429803830539390172192863012, 10.00722571186677189946782179772

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