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Properties

Label 2-1323-9.4-c1-0-34
Degree $2$
Conductor $1323$
Sign $-0.527 + 0.849i$
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  + (0.0341 + 0.0592i)2-s + (0.997 − 1.72i)4-s + (1.33 − 2.30i)5-s + 0.273·8-s + 0.182·10-s + (−0.799 − 1.38i)11-s + (2.62 − 4.54i)13-s + (−1.98 − 3.43i)16-s − 6.54·17-s − 1.90·19-s + (−2.65 − 4.60i)20-s + (0.0546 − 0.0946i)22-s + (−1.53 + 2.65i)23-s + (−1.04 − 1.81i)25-s + 0.359·26-s + ⋯
L(s)  = 1  + (0.0241 + 0.0418i)2-s + (0.498 − 0.864i)4-s + (0.595 − 1.03i)5-s + 0.0965·8-s + 0.0575·10-s + (−0.241 − 0.417i)11-s + (0.728 − 1.26i)13-s + (−0.496 − 0.859i)16-s − 1.58·17-s − 0.436·19-s + (−0.594 − 1.02i)20-s + (0.0116 − 0.0201i)22-s + (−0.319 + 0.554i)23-s + (−0.209 − 0.363i)25-s + 0.0704·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.527 + 0.849i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.527 + 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.527 + 0.849i$
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: \((2,\ 1323,\ (\ :1/2),\ -0.527 + 0.849i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.859486908\)
\(L(\frac12)\) \(\approx\) \(1.859486908\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0341 - 0.0592i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-1.33 + 2.30i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.799 + 1.38i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.62 + 4.54i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 6.54T + 17T^{2} \)
19 \( 1 + 1.90T + 19T^{2} \)
23 \( 1 + (1.53 - 2.65i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.19 - 5.53i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.35 + 5.81i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.22T + 37T^{2} \)
41 \( 1 + (3.69 - 6.40i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.63 - 9.75i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.89 + 3.29i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 8.89T + 53T^{2} \)
59 \( 1 + (-5.44 + 9.43i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.35 - 2.35i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.66 + 2.87i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.3T + 71T^{2} \)
73 \( 1 + 2.19T + 73T^{2} \)
79 \( 1 + (0.406 + 0.704i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.41 + 5.92i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 0.470T + 89T^{2} \)
97 \( 1 + (2.57 + 4.46i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.436889838280341856293317283792, −8.580021240047933755579915786181, −7.909578641119564075882989794897, −6.57018490261504023945039045911, −6.02518556736039711815128766438, −5.23153787014822193382755225809, −4.48870801611375159459338023559, −2.96607988305095617491573511252, −1.80117772073618637527381064705, −0.72780674646879059203803728670, 2.07834699021770493301709944499, 2.54407430740664187843431440622, 3.83467299011956412926778773272, 4.57546627459157694328432846571, 6.21092778599569330524677000028, 6.61420799877809325496807836085, 7.22048931840297209354714215657, 8.379507698439438859541657866954, 8.947207528061844677459448126189, 10.04745599049193404319586797752

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