↑

Properties

Label 2-1323-9.7-c1-0-18
Degree $2$
Conductor $1323$
Sign $0.949 - 0.313i$
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  + (0.649 − 1.12i)2-s + (0.155 + 0.268i)4-s + (1.76 + 3.05i)5-s + 3.00·8-s + 4.58·10-s + (0.589 − 1.02i)11-s + (1.61 + 2.78i)13-s + (1.64 − 2.84i)16-s + 4.90·17-s − 6.86·19-s + (−0.547 + 0.947i)20-s + (−0.765 − 1.32i)22-s + (−2.14 − 3.72i)23-s + (−3.71 + 6.43i)25-s + 4.18·26-s + ⋯
L(s)  = 1  + (0.459 − 0.796i)2-s + (0.0775 + 0.134i)4-s + (0.788 + 1.36i)5-s + 1.06·8-s + 1.44·10-s + (0.177 − 0.307i)11-s + (0.446 + 0.773i)13-s + (0.410 − 0.710i)16-s + 1.18·17-s − 1.57·19-s + (−0.122 + 0.211i)20-s + (−0.163 − 0.282i)22-s + (−0.448 − 0.776i)23-s + (−0.743 + 1.28i)25-s + 0.821·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.313i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.949 - 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.949 - 0.313i$
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: \((2,\ 1323,\ (\ :1/2),\ 0.949 - 0.313i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.810569678\)
\(L(\frac12)\) \(\approx\) \(2.810569678\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.649 + 1.12i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-1.76 - 3.05i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.589 + 1.02i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.61 - 2.78i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 4.90T + 17T^{2} \)
19 \( 1 + 6.86T + 19T^{2} \)
23 \( 1 + (2.14 + 3.72i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.36 - 2.35i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.960 - 1.66i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 9.76T + 37T^{2} \)
41 \( 1 + (-3.32 - 5.76i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.83 + 8.37i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.316 + 0.548i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 2.22T + 53T^{2} \)
59 \( 1 + (-4.10 - 7.11i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.82 + 8.36i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.66 + 4.61i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.27T + 71T^{2} \)
73 \( 1 - 1.03T + 73T^{2} \)
79 \( 1 + (0.502 - 0.869i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.65 + 6.33i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 12.0T + 89T^{2} \)
97 \( 1 + (5.46 - 9.46i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.12365954082263807505538011750, −8.977411829161882843943244613645, −8.037709593207836820626090230562, −6.96227625942905427859732940373, −6.48970693488913202548021605203, −5.50003440977681906406569758413, −4.18060455067583106659351645063, −3.42984094349620868078292796555, −2.51310927296469661661046943980, −1.71458725624069241307820091214, 1.08877070301752932631193591820, 2.07729987362697379990740806154, 3.87368083708265735379776206229, 4.74602249900649271323841030655, 5.68132106023384051653367685600, 5.85781030488455397229799323168, 7.01539900136143226642535924401, 7.993352133931369973918839895947, 8.593997181032409866306813422350, 9.619770146760492984323743555546

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