Properties

Label 2-1323-63.20-c1-0-19
Degree $2$
Conductor $1323$
Sign $-0.0164 + 0.999i$
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  + (−0.575 − 0.332i)2-s + (−0.779 − 1.34i)4-s + (0.0141 + 0.0245i)5-s + 2.36i·8-s − 0.0188i·10-s + (0.885 + 0.511i)11-s + (4.87 − 2.81i)13-s + (−0.773 + 1.33i)16-s − 5.67·17-s + 2.09i·19-s + (0.0220 − 0.0382i)20-s + (−0.339 − 0.588i)22-s + (6.28 − 3.63i)23-s + (2.49 − 4.32i)25-s − 3.74·26-s + ⋯
L(s)  = 1  + (−0.406 − 0.234i)2-s + (−0.389 − 0.674i)4-s + (0.00632 + 0.0109i)5-s + 0.835i·8-s − 0.00594i·10-s + (0.266 + 0.154i)11-s + (1.35 − 0.781i)13-s + (−0.193 + 0.334i)16-s − 1.37·17-s + 0.480i·19-s + (0.00493 − 0.00854i)20-s + (−0.0723 − 0.125i)22-s + (1.31 − 0.757i)23-s + (0.499 − 0.865i)25-s − 0.733·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0164 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0164 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.0164 + 0.999i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (440, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ -0.0164 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.128284401\)
\(L(\frac12)\) \(\approx\) \(1.128284401\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.575 + 0.332i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-0.0141 - 0.0245i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.885 - 0.511i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.87 + 2.81i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 5.67T + 17T^{2} \)
19 \( 1 - 2.09iT - 19T^{2} \)
23 \( 1 + (-6.28 + 3.63i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.52 - 2.03i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.87 - 1.65i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.47T + 37T^{2} \)
41 \( 1 + (-3.52 - 6.11i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.15 - 2.00i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.43 + 9.42i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 11.5iT - 53T^{2} \)
59 \( 1 + (3.01 + 5.21i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.05 + 1.18i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.38 + 11.0i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 7.93iT - 71T^{2} \)
73 \( 1 - 10.8iT - 73T^{2} \)
79 \( 1 + (-7.80 + 13.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.07 + 5.32i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 12.0T + 89T^{2} \)
97 \( 1 + (6.77 + 3.91i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.323592493911285512732468622660, −8.684300089859648772082377679554, −8.203364513085108914896300816293, −6.78542960974883978786349941796, −6.20228410395738239009294638331, −5.15798289944790200672457205480, −4.38759614816600542699697677899, −3.14822264261548290080165411500, −1.84399525073209025920299687466, −0.65143111929315892684068481776, 1.17768449821022030267058445654, 2.80130369310590349646026495295, 3.87320556911973987207657697587, 4.53911028786491796967484659918, 5.81219043556688864423041786709, 6.83195783117199431192093210407, 7.29510065150752134966850736543, 8.450609601705821968669880506606, 9.073082217491752043263787799320, 9.303344288453095608498241826825

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