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Properties

Label 2-1323-9.4-c1-0-28
Degree $2$
Conductor $1323$
Sign $0.384 + 0.923i$
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  + (0.335 + 0.580i)2-s + (0.775 − 1.34i)4-s + (0.712 − 1.23i)5-s + 2.38·8-s + 0.955·10-s + (−2.46 − 4.27i)11-s + (−1.37 + 2.38i)13-s + (−0.752 − 1.30i)16-s + 1.11·17-s + 4.01·19-s + (−1.10 − 1.91i)20-s + (1.65 − 2.86i)22-s + (2.71 − 4.70i)23-s + (1.48 + 2.57i)25-s − 1.84·26-s + ⋯
L(s)  = 1  + (0.236 + 0.410i)2-s + (0.387 − 0.671i)4-s + (0.318 − 0.551i)5-s + 0.841·8-s + 0.302·10-s + (−0.743 − 1.28i)11-s + (−0.381 + 0.661i)13-s + (−0.188 − 0.326i)16-s + 0.271·17-s + 0.921·19-s + (−0.247 − 0.427i)20-s + (0.352 − 0.610i)22-s + (0.566 − 0.981i)23-s + (0.296 + 0.514i)25-s − 0.362·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.384 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.384 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.384 + 0.923i$
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: \((2,\ 1323,\ (\ :1/2),\ 0.384 + 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.087238164\)
\(L(\frac12)\) \(\approx\) \(2.087238164\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.335 - 0.580i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-0.712 + 1.23i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.46 + 4.27i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.37 - 2.38i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 1.11T + 17T^{2} \)
19 \( 1 - 4.01T + 19T^{2} \)
23 \( 1 + (-2.71 + 4.70i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.40 + 5.89i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.25 - 2.17i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 1.41T + 37T^{2} \)
41 \( 1 + (0.124 - 0.215i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.498 + 0.863i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.73 + 8.20i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 0.820T + 53T^{2} \)
59 \( 1 + (3.29 - 5.70i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.0376 + 0.0651i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.29 + 10.9i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.0804T + 71T^{2} \)
73 \( 1 + 10.6T + 73T^{2} \)
79 \( 1 + (-0.922 - 1.59i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.23 - 12.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 13.5T + 89T^{2} \)
97 \( 1 + (-2.70 - 4.67i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.475698886962173962163410856601, −8.682190721831473428113497737436, −7.75658737888689966502663717451, −6.93071572430492252867443295523, −6.02360323471418126797571313141, −5.34804865356822255946542835571, −4.71996774078159882069813164195, −3.30709492089984880721866339847, −2.08022518567363947043887098534, −0.793940379652615915884085121590, 1.71046127065854867216236221022, 2.74922587687681224709025507764, 3.40101799079854517326521036450, 4.67582762899794319357365884075, 5.43316918948269709543580226462, 6.68711424602107103160820196463, 7.49981294086654317049123402015, 7.78178521291908012156913757463, 9.121229033769336596775077046454, 10.01046146471399842970080867071

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