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Properties

Label 2-1323-63.38-c1-0-29
Degree $2$
Conductor $1323$
Sign $-0.861 + 0.508i$
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  − 1.48i·2-s − 0.203·4-s + (0.154 − 0.267i)5-s − 2.66i·8-s + (−0.396 − 0.228i)10-s + (−2.73 + 1.58i)11-s + (3.00 − 1.73i)13-s − 4.36·16-s + (2.44 − 4.22i)17-s + (4.62 − 2.67i)19-s + (−0.0314 + 0.0544i)20-s + (2.34 + 4.06i)22-s + (−5.17 − 2.98i)23-s + (2.45 + 4.24i)25-s + (−2.57 − 4.45i)26-s + ⋯
L(s)  = 1  − 1.04i·2-s − 0.101·4-s + (0.0689 − 0.119i)5-s − 0.942i·8-s + (−0.125 − 0.0723i)10-s + (−0.825 + 0.476i)11-s + (0.833 − 0.481i)13-s − 1.09·16-s + (0.592 − 1.02i)17-s + (1.06 − 0.612i)19-s + (−0.00702 + 0.0121i)20-s + (0.500 + 0.866i)22-s + (−1.07 − 0.622i)23-s + (0.490 + 0.849i)25-s + (−0.504 − 0.874i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.861 + 0.508i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.861 + 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.861 + 0.508i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ -0.861 + 0.508i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.668713523\)
\(L(\frac12)\) \(\approx\) \(1.668713523\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.48iT - 2T^{2} \)
5 \( 1 + (-0.154 + 0.267i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.73 - 1.58i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.00 + 1.73i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.44 + 4.22i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.62 + 2.67i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.17 + 2.98i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.70 + 1.56i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 7.52iT - 31T^{2} \)
37 \( 1 + (5.92 + 10.2i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.58 + 4.48i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.75 + 4.76i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 8.46T + 47T^{2} \)
53 \( 1 + (0.0740 + 0.0427i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 2.08T + 59T^{2} \)
61 \( 1 + 5.42iT - 61T^{2} \)
67 \( 1 + 0.110T + 67T^{2} \)
71 \( 1 - 7.78iT - 71T^{2} \)
73 \( 1 + (8.32 + 4.80i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 5.13T + 79T^{2} \)
83 \( 1 + (-4.42 + 7.66i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.936 - 1.62i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (10.9 + 6.34i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.451880764457602445764912741810, −8.780209425240530442330806894897, −7.51873702194332303026125329678, −7.07919049509013130071563140252, −5.75607888353584585665595921300, −5.01129604381553607011959700675, −3.75131158937659299403034444985, −2.98579496209273933063182368763, −1.98306966036975292655755886006, −0.69003189293825921586900213711, 1.60975329536825823153588055147, 2.95328930075473802393408859207, 4.05534597780292782644335419850, 5.32425027786411831218736875671, 5.93866113104438018353266943603, 6.53861560779605379186533006696, 7.72805568679030685620365293203, 8.008766285597220250472965862680, 8.863178490049323522057260616144, 9.962727968024243317628258888544

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