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Properties

Label 2-1323-63.25-c1-0-26
Degree $2$
Conductor $1323$
Sign $-0.678 + 0.734i$
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  − 1.34·2-s − 0.184·4-s + (1.26 − 2.19i)5-s + 2.94·8-s + (−1.70 + 2.95i)10-s + (0.233 + 0.405i)11-s + (−2.91 − 5.04i)13-s − 3.59·16-s + (1.93 − 3.35i)17-s + (1.09 + 1.89i)19-s + (−0.233 + 0.405i)20-s + (−0.315 − 0.545i)22-s + (−0.0530 + 0.0918i)23-s + (−0.705 − 1.22i)25-s + (3.92 + 6.79i)26-s + ⋯
L(s)  = 1  − 0.952·2-s − 0.0923·4-s + (0.566 − 0.980i)5-s + 1.04·8-s + (−0.539 + 0.934i)10-s + (0.0705 + 0.122i)11-s + (−0.807 − 1.39i)13-s − 0.899·16-s + (0.470 − 0.814i)17-s + (0.250 + 0.434i)19-s + (−0.0523 + 0.0906i)20-s + (−0.0672 − 0.116i)22-s + (−0.0110 + 0.0191i)23-s + (−0.141 − 0.244i)25-s + (0.769 + 1.33i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.678 + 0.734i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.678 + 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.678 + 0.734i$
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: \((2,\ 1323,\ (\ :1/2),\ -0.678 + 0.734i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7060579436\)
\(L(\frac12)\) \(\approx\) \(0.7060579436\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.34T + 2T^{2} \)
5 \( 1 + (-1.26 + 2.19i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.233 - 0.405i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.91 + 5.04i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.93 + 3.35i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.09 - 1.89i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.0530 - 0.0918i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.39 + 7.60i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 7.68T + 31T^{2} \)
37 \( 1 + (-3.84 - 6.65i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.11 + 1.92i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.613 - 1.06i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 5.33T + 47T^{2} \)
53 \( 1 + (0.358 - 0.620i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 0.736T + 59T^{2} \)
61 \( 1 - 0.958T + 61T^{2} \)
67 \( 1 + 9.63T + 67T^{2} \)
71 \( 1 + 13.2T + 71T^{2} \)
73 \( 1 + (-5.13 + 8.89i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 12.6T + 79T^{2} \)
83 \( 1 + (1.36 - 2.36i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.05 + 7.02i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.80 + 11.7i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.386736093297137183208907528362, −8.632920556891456090505401288560, −7.85003893481137941710526620717, −7.30465567411923526402838435651, −5.84431803504516293134210128832, −5.15647210108345494806655841704, −4.37733122918936761059266255226, −2.88553098963996835285390837704, −1.50691906677424877124384919067, −0.43651933964689686108467198923, 1.49376423775150891738678551928, 2.53019707865550335147930614360, 3.84639525545306640528622805239, 4.86380285581696211941548942749, 5.96580661870092822463994557396, 7.01159818697596216983890284844, 7.34765527993913817589122741752, 8.546419604787823953684772182599, 9.189273954470415373296145774940, 9.826699419650117911064212539462

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