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Properties

Label 2-1323-9.4-c1-0-6
Degree $2$
Conductor $1323$
Sign $-0.0644 + 0.997i$
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  + (1.23 + 2.13i)2-s + (−2.02 + 3.51i)4-s + (−1.82 + 3.16i)5-s − 5.05·8-s − 9.00·10-s + (0.203 + 0.351i)11-s + (0.243 − 0.421i)13-s + (−2.16 − 3.74i)16-s − 4.85·17-s − 1.97·19-s + (−7.41 − 12.8i)20-s + (−0.5 + 0.866i)22-s + (2.32 − 4.02i)23-s + (−4.19 − 7.25i)25-s + 1.19·26-s + ⋯
L(s)  = 1  + (0.869 + 1.50i)2-s + (−1.01 + 1.75i)4-s + (−0.817 + 1.41i)5-s − 1.78·8-s − 2.84·10-s + (0.0612 + 0.106i)11-s + (0.0675 − 0.116i)13-s + (−0.540 − 0.936i)16-s − 1.17·17-s − 0.452·19-s + (−1.65 − 2.87i)20-s + (−0.106 + 0.184i)22-s + (0.484 − 0.839i)23-s + (−0.838 − 1.45i)25-s + 0.234·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0644 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0644 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.0644 + 0.997i$
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: \((2,\ 1323,\ (\ :1/2),\ -0.0644 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.369619651\)
\(L(\frac12)\) \(\approx\) \(1.369619651\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.23 - 2.13i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (1.82 - 3.16i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.203 - 0.351i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.243 + 0.421i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 4.85T + 17T^{2} \)
19 \( 1 + 1.97T + 19T^{2} \)
23 \( 1 + (-2.32 + 4.02i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.82 - 6.62i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.51 - 6.08i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2.32T + 37T^{2} \)
41 \( 1 + (-3.75 + 6.50i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.16 - 2.01i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.15 + 5.47i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 3.56T + 53T^{2} \)
59 \( 1 + (-3.05 + 5.29i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.01 - 6.95i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.80 - 3.11i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.46T + 71T^{2} \)
73 \( 1 - 1.97T + 73T^{2} \)
79 \( 1 + (4.08 + 7.06i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.08 - 10.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 14.8T + 89T^{2} \)
97 \( 1 + (-4.74 - 8.21i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.47499898846932857839595511727, −8.934263840082250891388201377252, −8.346811522525750081785554718805, −7.37688759832109467666402904995, −6.83959913929082937267750909238, −6.44458472128611692848202662911, −5.28762519687642589393601235113, −4.35505954593621784695691165349, −3.61218279556082011461441451257, −2.63556187950415170784297958086, 0.42616091194259896210819977014, 1.58358840935837128510513700338, 2.75110621242514839724020129125, 4.06261680819137130365672879211, 4.32883822940685016548528144318, 5.19969379137110303355046431442, 6.14488014562039262571551375340, 7.58272803927237863260109453982, 8.508475249586462490712775582924, 9.241974903670556134677661175462

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