↑

Properties

Label 2-1323-9.7-c1-0-35
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

Related objects

Downloads

Learn more

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: $\chi_{1323} (883, \cdot )$
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 \( 1 + (-4.74 + 8.21i)T + (-48.5 - 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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