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Properties

Label 2-1323-63.59-c1-0-6
Degree $2$
Conductor $1323$
Sign $0.0635 - 0.997i$
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  + (0.105 + 0.0611i)2-s + (−0.992 − 1.71i)4-s − 0.529·5-s − 0.487i·8-s + (−0.0560 − 0.0323i)10-s + 4.20i·11-s + (−1.74 − 1.00i)13-s + (−1.95 + 3.38i)16-s + (2.19 − 3.79i)17-s + (−4.54 + 2.62i)19-s + (0.525 + 0.910i)20-s + (−0.257 + 0.445i)22-s + 6.27i·23-s − 4.71·25-s + (−0.123 − 0.213i)26-s + ⋯
L(s)  = 1  + (0.0749 + 0.0432i)2-s + (−0.496 − 0.859i)4-s − 0.236·5-s − 0.172i·8-s + (−0.0177 − 0.0102i)10-s + 1.26i·11-s + (−0.484 − 0.279i)13-s + (−0.488 + 0.846i)16-s + (0.532 − 0.921i)17-s + (−1.04 + 0.601i)19-s + (0.117 + 0.203i)20-s + (−0.0548 + 0.0949i)22-s + 1.30i·23-s − 0.943·25-s + (−0.0242 − 0.0419i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0635 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0635 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.0635 - 0.997i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (962, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ 0.0635 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7961456924\)
\(L(\frac12)\) \(\approx\) \(0.7961456924\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.105 - 0.0611i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + 0.529T + 5T^{2} \)
11 \( 1 - 4.20iT - 11T^{2} \)
13 \( 1 + (1.74 + 1.00i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.19 + 3.79i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.54 - 2.62i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 6.27iT - 23T^{2} \)
29 \( 1 + (-7.27 + 4.20i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.03 - 0.595i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.61 - 2.79i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.0994 + 0.172i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.96 - 6.86i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.98 - 8.63i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.65 - 2.10i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.71 - 11.6i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (11.3 + 6.55i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.29 - 5.69i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.50iT - 71T^{2} \)
73 \( 1 + (-4.86 - 2.80i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.286 - 0.495i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.42 + 9.39i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-6.43 - 11.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.493 + 0.285i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.822374304080883989527009730187, −9.298768887068039703433624092060, −8.111967509600669927954093750366, −7.44668001219216109375753561590, −6.44720283762218010694665859492, −5.58885741684692672187543611301, −4.71639657006925010367978503402, −4.06056631889958115962458750695, −2.57719076106772562993372270982, −1.32622610427092126020511235831, 0.34180931942456184879886322920, 2.32389259975456805028065747490, 3.40015494951497328731226364455, 4.15117500823409137576484766971, 5.06838571237407014151559524275, 6.18213916834387926597512268751, 7.00788678884004330459268997282, 8.131806320857075135372391715151, 8.452824505020598274127621848377, 9.210128179145594379524645088291

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