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Properties

Label 2-1323-63.41-c1-0-15
Degree $2$
Conductor $1323$
Sign $0.946 - 0.322i$
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  + (1.30 − 0.755i)2-s + (0.140 − 0.242i)4-s + (0.387 − 0.671i)5-s + 2.59i·8-s − 1.17i·10-s + (−3.32 + 1.92i)11-s + (2.54 + 1.46i)13-s + (2.24 + 3.88i)16-s + 5.39·17-s + 0.434i·19-s + (−0.108 − 0.188i)20-s + (−2.90 + 5.02i)22-s + (−0.0482 − 0.0278i)23-s + (2.19 + 3.80i)25-s + 4.43·26-s + ⋯
L(s)  = 1  + (0.924 − 0.533i)2-s + (0.0700 − 0.121i)4-s + (0.173 − 0.300i)5-s + 0.918i·8-s − 0.370i·10-s + (−1.00 + 0.579i)11-s + (0.705 + 0.407i)13-s + (0.560 + 0.970i)16-s + 1.30·17-s + 0.0997i·19-s + (−0.0243 − 0.0421i)20-s + (−0.618 + 1.07i)22-s + (−0.0100 − 0.00580i)23-s + (0.439 + 0.761i)25-s + 0.869·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 - 0.322i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.946 - 0.322i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.946 - 0.322i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ 0.946 - 0.322i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.601900109\)
\(L(\frac12)\) \(\approx\) \(2.601900109\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.30 + 0.755i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (-0.387 + 0.671i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.32 - 1.92i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.54 - 1.46i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 5.39T + 17T^{2} \)
19 \( 1 - 0.434iT - 19T^{2} \)
23 \( 1 + (0.0482 + 0.0278i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.187 + 0.108i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.67 - 3.27i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 6.29T + 37T^{2} \)
41 \( 1 + (-3.78 + 6.56i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.42 - 11.1i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.482 - 0.836i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 7.46iT - 53T^{2} \)
59 \( 1 + (1.56 - 2.70i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.01 - 1.74i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.10 + 3.63i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.50iT - 71T^{2} \)
73 \( 1 + 8.15iT - 73T^{2} \)
79 \( 1 + (-2.48 - 4.29i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.31 + 7.46i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 15.6T + 89T^{2} \)
97 \( 1 + (1.24 - 0.716i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.820190647177398666219667563766, −8.834118807655393889480967434515, −8.072684748225980311756523726013, −7.28352900262384909927580246056, −6.01164464959458122605870654869, −5.26144862024979778011501527410, −4.57184926295203815472586468670, −3.54913152693074962151419867222, −2.71195474888217828530905690864, −1.48830590110788396782177042474, 0.876268404524141113093470062423, 2.73823894941580358654853861647, 3.57109107368540441078633196763, 4.61806008184864640364500136424, 5.60055441033466643433441397189, 5.95170795367845750108141679953, 6.95258712611543705533003691052, 7.82814855964577377427097945683, 8.595963121756123052353017420463, 9.768906967538032589443408260398

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