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Properties

Label 2-1323-9.4-c1-0-15
Degree $2$
Conductor $1323$
Sign $0.766 + 0.642i$
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 + (−0.673 + 1.16i)5-s − 2.83·8-s + 1.18·10-s + (0.826 + 1.43i)11-s + (−1.68 + 2.91i)13-s + (0.0209 + 0.0362i)16-s + 0.467·17-s + 3.22·19-s + (0.826 + 1.43i)20-s + (0.726 − 1.25i)22-s + (4.47 − 7.74i)23-s + (1.59 + 2.75i)25-s + 2.96·26-s + ⋯
L(s)  = 1  + (−0.310 − 0.538i)2-s + (0.306 − 0.531i)4-s + (−0.301 + 0.521i)5-s − 1.00·8-s + 0.374·10-s + (0.249 + 0.431i)11-s + (−0.467 + 0.809i)13-s + (0.00523 + 0.00906i)16-s + 0.113·17-s + 0.740·19-s + (0.184 + 0.320i)20-s + (0.154 − 0.268i)22-s + (0.932 − 1.61i)23-s + (0.318 + 0.551i)25-s + 0.581·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.766 + 0.642i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (442, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ 0.766 + 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.406961320\)
\(L(\frac12)\) \(\approx\) \(1.406961320\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.673 - 1.16i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.826 - 1.43i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.68 - 2.91i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 0.467T + 17T^{2} \)
19 \( 1 - 3.22T + 19T^{2} \)
23 \( 1 + (-4.47 + 7.74i)T + (-11.5 - 19.9i)T^{2} \)
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 \( 1 - 9.23T + 37T^{2} \)
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 \( 1 - 0.573T + 53T^{2} \)
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 \( 1 + 2.04T + 73T^{2} \)
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 \( 1 - 9.08T + 89T^{2} \)
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

−9.659356406940470286261841822272, −9.013049541586565108295733789619, −7.954886175212527276481583036664, −6.84846074026995778255945267574, −6.55251198123408929477760616592, −5.29031188161482216164042303223, −4.37253054214197540511969216244, −3.07201088240351611640967905629, −2.28902996215265095176585872028, −0.944908187501929559442072601944, 0.903057635039349287963675294139, 2.73743623672108627455712007396, 3.49853898510576492463149524299, 4.74002776479727225896270540495, 5.66111955855991064287902722142, 6.54602984076486678745963817595, 7.49940289890916827259531349597, 7.975161783327786269724716950848, 8.775190624391324665651795627614, 9.471379902961234776499576603672

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