Properties

Label 2-1323-63.20-c1-0-21
Degree $2$
Conductor $1323$
Sign $0.291 + 0.956i$
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  + (−0.105 − 0.0611i)2-s + (−0.992 − 1.71i)4-s + (−0.264 − 0.458i)5-s + 0.487i·8-s + 0.0647i·10-s + (3.64 + 2.10i)11-s + (1.74 − 1.00i)13-s + (−1.95 + 3.38i)16-s + 4.38·17-s − 5.24i·19-s + (−0.525 + 0.910i)20-s + (−0.257 − 0.445i)22-s + (−5.43 + 3.13i)23-s + (2.35 − 4.08i)25-s − 0.246·26-s + ⋯
L(s)  = 1  + (−0.0749 − 0.0432i)2-s + (−0.496 − 0.859i)4-s + (−0.118 − 0.205i)5-s + 0.172i·8-s + 0.0204i·10-s + (1.09 + 0.633i)11-s + (0.484 − 0.279i)13-s + (−0.488 + 0.846i)16-s + 1.06·17-s − 1.20i·19-s + (−0.117 + 0.203i)20-s + (−0.0548 − 0.0949i)22-s + (−1.13 + 0.654i)23-s + (0.471 − 0.817i)25-s − 0.0484·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.291 + 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.291 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.291 + 0.956i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (440, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ 0.291 + 0.956i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.441925391\)
\(L(\frac12)\) \(\approx\) \(1.441925391\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.105 + 0.0611i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (0.264 + 0.458i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3.64 - 2.10i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.74 + 1.00i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 4.38T + 17T^{2} \)
19 \( 1 + 5.24iT - 19T^{2} \)
23 \( 1 + (5.43 - 3.13i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-7.27 - 4.20i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.03 - 0.595i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 3.23T + 37T^{2} \)
41 \( 1 + (0.0994 + 0.172i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.96 + 6.86i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.98 + 8.63i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 4.21iT - 53T^{2} \)
59 \( 1 + (6.71 + 11.6i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (11.3 + 6.55i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.29 - 5.69i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.50iT - 71T^{2} \)
73 \( 1 + 5.61iT - 73T^{2} \)
79 \( 1 + (0.286 - 0.495i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.42 + 9.39i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 12.8T + 89T^{2} \)
97 \( 1 + (0.493 + 0.285i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.358557968766731845763276505169, −8.899502881747630379754279546525, −7.969239115028263744524011046078, −6.87331458522389257116768464964, −6.14872240266310933596003248655, −5.18467510994006877493549515273, −4.44594093938565305098977334675, −3.43929726142377184511071722764, −1.86793819239261347824714592913, −0.76265963553010480818212108176, 1.20543459476724308925110979509, 2.90978575823178572331755255584, 3.76952725716523415317636945946, 4.39697854464597927179241686722, 5.79062154230297953875273552059, 6.47006636748589106827961407888, 7.57123210001549462839533535827, 8.171281569511352258408167635735, 8.903084271218048268985632471618, 9.655908115359556209794154485266

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