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Properties

Label 2-1323-63.25-c1-0-1
Degree $2$
Conductor $1323$
Sign $-0.592 - 0.805i$
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  + 0.239·2-s − 1.94·4-s + (0.590 − 1.02i)5-s − 0.942·8-s + (0.141 − 0.244i)10-s + (−1.85 − 3.20i)11-s + (−0.5 − 0.866i)13-s + 3.66·16-s + (−3.47 + 6.01i)17-s + (−0.971 − 1.68i)19-s + (−1.14 + 1.98i)20-s + (−0.442 − 0.766i)22-s + (−2.80 + 4.85i)23-s + (1.80 + 3.12i)25-s + (−0.119 − 0.207i)26-s + ⋯
L(s)  = 1  + 0.169·2-s − 0.971·4-s + (0.264 − 0.457i)5-s − 0.333·8-s + (0.0446 − 0.0774i)10-s + (−0.558 − 0.967i)11-s + (−0.138 − 0.240i)13-s + 0.915·16-s + (−0.841 + 1.45i)17-s + (−0.222 − 0.385i)19-s + (−0.256 + 0.444i)20-s + (−0.0944 − 0.163i)22-s + (−0.584 + 1.01i)23-s + (0.360 + 0.624i)25-s + (−0.0234 − 0.0406i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.592 - 0.805i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.592 - 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.592 - 0.805i$
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: \((2,\ 1323,\ (\ :1/2),\ -0.592 - 0.805i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3996807516\)
\(L(\frac12)\) \(\approx\) \(0.3996807516\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.239T + 2T^{2} \)
5 \( 1 + (-0.590 + 1.02i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.85 + 3.20i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.47 - 6.01i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.971 + 1.68i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.80 - 4.85i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.119 + 0.207i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.66T + 31T^{2} \)
37 \( 1 + (-4.77 - 8.26i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.09 + 8.81i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.11 - 1.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 5.82T + 47T^{2} \)
53 \( 1 + (5.80 - 10.0i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 2.60T + 59T^{2} \)
61 \( 1 + 7.60T + 61T^{2} \)
67 \( 1 - 3.50T + 67T^{2} \)
71 \( 1 + 8.60T + 71T^{2} \)
73 \( 1 + (7.57 - 13.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 7.37T + 79T^{2} \)
83 \( 1 + (3.47 - 6.01i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.37 - 2.37i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.58 - 6.20i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.848092085249642689408782925222, −8.970273833204323009096298503487, −8.446442472321904983377300145714, −7.74823409704684600875812991935, −6.36123504694371624572985624487, −5.65479339761280156888499453767, −4.87446421233761338316934978290, −3.97201955753257460805180790472, −2.99636386068624116479470122842, −1.40815270687679638559048349687, 0.16271053730768304552844842518, 2.12268439234079227724664535969, 3.12197714873859518006820076239, 4.54026875418306292203550520744, 4.76241308872283839482829837903, 6.02381670397544633572899976735, 6.83757247004473846969537734841, 7.76860926900100700568155963327, 8.588970060985863520092918046552, 9.493726826968732226869271332341

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