Properties

Label 2-1323-63.4-c1-0-24
Degree $2$
Conductor $1323$
Sign $-0.0477 + 0.998i$
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  + (−0.439 + 0.761i)2-s + (0.613 + 1.06i)4-s − 1.34·5-s − 2.83·8-s + (0.592 − 1.02i)10-s − 1.65·11-s + (1.68 − 2.91i)13-s + (0.0209 − 0.0362i)16-s + (0.233 − 0.405i)17-s + (1.61 + 2.79i)19-s + (−0.826 − 1.43i)20-s + (0.726 − 1.25i)22-s − 8.94·23-s − 3.18·25-s + (1.48 + 2.56i)26-s + ⋯
L(s)  = 1  + (−0.310 + 0.538i)2-s + (0.306 + 0.531i)4-s − 0.602·5-s − 1.00·8-s + (0.187 − 0.324i)10-s − 0.498·11-s + (0.467 − 0.809i)13-s + (0.00523 − 0.00906i)16-s + (0.0567 − 0.0982i)17-s + (0.370 + 0.641i)19-s + (−0.184 − 0.320i)20-s + (0.154 − 0.268i)22-s − 1.86·23-s − 0.636·25-s + (0.290 + 0.503i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0477 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0477 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.0477 + 0.998i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ -0.0477 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2447966629\)
\(L(\frac12)\) \(\approx\) \(0.2447966629\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.439 - 0.761i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + 1.34T + 5T^{2} \)
11 \( 1 + 1.65T + 11T^{2} \)
13 \( 1 + (-1.68 + 2.91i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.233 + 0.405i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.61 - 2.79i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 8.94T + 23T^{2} \)
29 \( 1 + (-3.13 - 5.42i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.61 + 7.99i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.61 + 7.99i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.70 + 2.95i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.20 - 3.82i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.67 + 8.10i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.286 - 0.497i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.19 + 9.00i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.81 - 6.61i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.298 + 0.516i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 0.554T + 71T^{2} \)
73 \( 1 + (1.02 - 1.77i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.20 + 2.08i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.52 + 13.0i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.54 - 7.86i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.949 - 1.64i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.223570309217767342256618661600, −8.328921301536463209737446378188, −7.77687579406969938294000501332, −7.30060332815684396997643896721, −6.08288844921793090447818228630, −5.54814605798431616837093404881, −4.02886665233165215115019340638, −3.39975538671496479765347852507, −2.15997975621734062114748632843, −0.10830378920150158826795114865, 1.44887806423054586927080017868, 2.54142849841154333257837602021, 3.64784947781896444673392192901, 4.66503261548998684958166996016, 5.78645829763341735342439504258, 6.49574745541288859486302094520, 7.47635529462702219574279226575, 8.334745900316526553194461704228, 9.153462492260880327898135567550, 9.957738263539273347098491800375

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