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Properties

Label 2-1323-63.38-c1-0-13
Degree $2$
Conductor $1323$
Sign $0.310 - 0.950i$
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  + 0.122i·2-s + 1.98·4-s + (−0.264 + 0.458i)5-s + 0.487i·8-s + (−0.0560 − 0.0323i)10-s + (−3.64 + 2.10i)11-s + (1.74 − 1.00i)13-s + 3.91·16-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 + (5.43 + 3.13i)23-s + (2.35 + 4.08i)25-s + (0.123 + 0.213i)26-s + ⋯
L(s)  = 1  + 0.0865i·2-s + 0.992·4-s + (−0.118 + 0.205i)5-s + 0.172i·8-s + (−0.0177 − 0.0102i)10-s + (−1.09 + 0.633i)11-s + (0.484 − 0.279i)13-s + 0.977·16-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 + (1.13 + 0.654i)23-s + (0.471 + 0.817i)25-s + (0.0242 + 0.0419i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.310 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.310 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.310 - 0.950i$
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: \((2,\ 1323,\ (\ :1/2),\ 0.310 - 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.844794108\)
\(L(\frac12)\) \(\approx\) \(1.844794108\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.122iT - 2T^{2} \)
5 \( 1 + (0.264 - 0.458i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.64 - 2.10i)T + (5.5 - 9.52i)T^{2} \)
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 \( 1 + (-5.43 - 3.13i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-7.27 - 4.20i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.19iT - 31T^{2} \)
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 \( 1 + 9.97T + 47T^{2} \)
53 \( 1 + (3.65 + 2.10i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 13.4T + 59T^{2} \)
61 \( 1 - 13.1iT - 61T^{2} \)
67 \( 1 + 6.58T + 67T^{2} \)
71 \( 1 + 8.50iT - 71T^{2} \)
73 \( 1 + (-4.86 - 2.80i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 0.572T + 79T^{2} \)
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

−10.11967360211260506455900925326, −8.785219052260771077045530291892, −8.111418921728846158123604694372, −7.26445182353466276222017986918, −6.60869148063559315815463868234, −5.73961756754339881414607746850, −4.81092158731867556720935788026, −3.53215974500869156925861418616, −2.61606666010297532875815303689, −1.54252830425411841215694674605, 0.74958101896502099700472357145, 2.40831181383163848691650242653, 2.95671789504805843482240560106, 4.39172091083855443083282470310, 5.24056459757554536663009331540, 6.44994649320466551310535868476, 6.76128499130003942092976456418, 7.995856630716207116227576987674, 8.463061689715735618389479176437, 9.498748744273191568581468574567

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