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Properties

Label 2-1323-63.41-c1-0-1
Degree $2$
Conductor $1323$
Sign $-0.967 + 0.251i$
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  + (0.850 − 0.490i)2-s + (−0.518 + 0.897i)4-s + (−0.940 + 1.62i)5-s + 2.98i·8-s + 1.84i·10-s + (−3.54 + 2.04i)11-s + (−3.51 − 2.02i)13-s + (0.426 + 0.738i)16-s + 1.62·17-s − 8.12i·19-s + (−0.974 − 1.68i)20-s + (−2.00 + 3.47i)22-s + (−3.73 − 2.15i)23-s + (0.730 + 1.26i)25-s − 3.98·26-s + ⋯
L(s)  = 1  + (0.601 − 0.347i)2-s + (−0.259 + 0.448i)4-s + (−0.420 + 0.728i)5-s + 1.05i·8-s + 0.583i·10-s + (−1.06 + 0.616i)11-s + (−0.974 − 0.562i)13-s + (0.106 + 0.184i)16-s + 0.393·17-s − 1.86i·19-s + (−0.217 − 0.377i)20-s + (−0.427 + 0.740i)22-s + (−0.778 − 0.449i)23-s + (0.146 + 0.253i)25-s − 0.781·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 + 0.251i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.967 + 0.251i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.967 + 0.251i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ -0.967 + 0.251i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1622119918\)
\(L(\frac12)\) \(\approx\) \(0.1622119918\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.850 + 0.490i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (0.940 - 1.62i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.54 - 2.04i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.51 + 2.02i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 1.62T + 17T^{2} \)
19 \( 1 + 8.12iT - 19T^{2} \)
23 \( 1 + (3.73 + 2.15i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.542 + 0.313i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.69 + 2.13i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 7.94T + 37T^{2} \)
41 \( 1 + (-0.912 + 1.57i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.53 + 6.12i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.96 + 6.87i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 8.37iT - 53T^{2} \)
59 \( 1 + (4.08 - 7.07i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.24 - 1.87i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.26 - 10.8i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 14.4iT - 71T^{2} \)
73 \( 1 + 3.78iT - 73T^{2} \)
79 \( 1 + (4.18 + 7.24i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.38 - 7.59i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 9.80T + 89T^{2} \)
97 \( 1 + (11.4 - 6.61i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.24854361666843426176818407333, −9.312481622460637962724221152245, −8.310113621018298779918274577360, −7.49348523019330042617093338291, −7.08991587877348837846948594736, −5.61858838635616739682193198164, −4.87447466624258008989113032904, −4.07152674717882238693980658624, −2.84745264212468265367650836429, −2.49903622845602085677375763763, 0.05288434654325812159899382224, 1.59533522486719982502028821920, 3.22331093668236487126522722256, 4.25600090318856072800369700733, 4.96675446176582371564117979785, 5.71245659558634620397087653925, 6.45886357991807399179193251886, 7.78766589723883723675567124141, 8.080476360977051548229315407382, 9.336271051692618404716712183599

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