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Properties

Label 2-1323-63.25-c1-0-33
Degree $2$
Conductor $1323$
Sign $0.747 + 0.663i$
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  + 2.38·2-s + 3.69·4-s + (1.46 − 2.52i)5-s + 4.05·8-s + (3.48 − 6.03i)10-s + (−0.676 − 1.17i)11-s + (0.733 + 1.26i)13-s + 2.27·16-s + (1.65 − 2.86i)17-s + (1.10 + 1.91i)19-s + (5.39 − 9.35i)20-s + (−1.61 − 2.79i)22-s + (1.31 − 2.27i)23-s + (−1.76 − 3.05i)25-s + (1.74 + 3.03i)26-s + ⋯
L(s)  = 1  + 1.68·2-s + 1.84·4-s + (0.653 − 1.13i)5-s + 1.43·8-s + (1.10 − 1.90i)10-s + (−0.204 − 0.353i)11-s + (0.203 + 0.352i)13-s + 0.568·16-s + (0.401 − 0.695i)17-s + (0.253 + 0.438i)19-s + (1.20 − 2.09i)20-s + (−0.344 − 0.596i)22-s + (0.274 − 0.474i)23-s + (−0.353 − 0.611i)25-s + (0.343 + 0.594i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.747 + 0.663i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.747 + 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.747 + 0.663i$
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: \((2,\ 1323,\ (\ :1/2),\ 0.747 + 0.663i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.890322082\)
\(L(\frac12)\) \(\approx\) \(4.890322082\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.38T + 2T^{2} \)
5 \( 1 + (-1.46 + 2.52i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.676 + 1.17i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.733 - 1.26i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.65 + 2.86i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.10 - 1.91i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.31 + 2.27i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.521 - 0.903i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.27T + 31T^{2} \)
37 \( 1 + (-5.43 - 9.41i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.904 + 1.56i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.17 - 3.76i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 3.97T + 47T^{2} \)
53 \( 1 + (-3.22 + 5.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 12.2T + 59T^{2} \)
61 \( 1 + 0.559T + 61T^{2} \)
67 \( 1 - 12.8T + 67T^{2} \)
71 \( 1 + 12.9T + 71T^{2} \)
73 \( 1 + (5.22 - 9.05i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 0.767T + 79T^{2} \)
83 \( 1 + (0.983 - 1.70i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-3.20 - 5.54i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.14 + 7.17i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.523548685908559441368909017227, −8.773733762021038356731958087305, −7.74673582163758599042392524634, −6.67043293262754569070531346060, −5.89677627513529099639385408149, −5.18249648802722584640882290212, −4.63658189273941780309465205325, −3.59080528112370906766423374274, −2.57344883179791376718157268520, −1.32237887330186119731725375770, 1.94684372126581882324238709076, 2.85563572801393743926693329118, 3.60082746443330472463175323661, 4.61132549685255244998003001976, 5.70437655198526131688388929002, 6.03040032010998208769594174772, 7.04572978487890701765045246506, 7.58436850993085972315042509610, 9.021346524766586259315363657894, 10.03419549722815090238794609310

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