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Properties

Label 2-1323-9.4-c1-0-32
Degree $2$
Conductor $1323$
Sign $-0.841 + 0.539i$
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  + (−0.247 − 0.429i)2-s + (0.877 − 1.51i)4-s + (1.84 − 3.19i)5-s − 1.86·8-s − 1.83·10-s + (−0.446 − 0.772i)11-s + (−0.598 + 1.03i)13-s + (−1.29 − 2.23i)16-s + 0.249·17-s + 2.80·19-s + (−3.23 − 5.60i)20-s + (−0.221 + 0.383i)22-s + (1.23 − 2.14i)23-s + (−4.31 − 7.47i)25-s + 0.593·26-s + ⋯
L(s)  = 1  + (−0.175 − 0.303i)2-s + (0.438 − 0.759i)4-s + (0.825 − 1.43i)5-s − 0.658·8-s − 0.579·10-s + (−0.134 − 0.233i)11-s + (−0.165 + 0.287i)13-s + (−0.323 − 0.559i)16-s + 0.0606·17-s + 0.644·19-s + (−0.724 − 1.25i)20-s + (−0.0471 + 0.0817i)22-s + (0.258 − 0.447i)23-s + (−0.863 − 1.49i)25-s + 0.116·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.841 + 0.539i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.841 + 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.841 + 0.539i$
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: \((2,\ 1323,\ (\ :1/2),\ -0.841 + 0.539i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.736453815\)
\(L(\frac12)\) \(\approx\) \(1.736453815\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.247 + 0.429i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-1.84 + 3.19i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.446 + 0.772i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.598 - 1.03i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 0.249T + 17T^{2} \)
19 \( 1 - 2.80T + 19T^{2} \)
23 \( 1 + (-1.23 + 2.14i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.07 + 3.58i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.79 + 3.10i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.73T + 37T^{2} \)
41 \( 1 + (2.39 - 4.14i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.98 + 8.64i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.08 - 8.81i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 9.88T + 53T^{2} \)
59 \( 1 + (0.906 - 1.56i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.40 - 9.35i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.514 - 0.891i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.94T + 71T^{2} \)
73 \( 1 + 1.83T + 73T^{2} \)
79 \( 1 + (-0.899 - 1.55i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.16 - 10.6i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 2.40T + 89T^{2} \)
97 \( 1 + (5.52 + 9.56i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.479836882007481941305729611691, −8.751621706887782006751906787716, −7.83292216885667864700498707532, −6.63393456687661820688387494049, −5.80855653415181875511900011683, −5.22835250073408756055154499499, −4.32655778981553825276412871115, −2.72560135898294259269370286451, −1.70560738680489533992696614508, −0.74180193776828483859916356895, 1.95011694088265778649428404670, 2.93953088624929289907387728642, 3.52332301489832171316167343631, 5.10974088976400154308248342378, 6.10345509263089118589779515809, 6.78553947061436273875020493500, 7.38093341065729580026908899926, 8.134117690545274684537572739129, 9.249267974181079178881181953878, 9.923298378483550222664662040396

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