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Properties

Label 2-1323-7.3-c0-0-0
Degree $2$
Conductor $1323$
Sign $0.444 + 0.895i$
Analytic cond. $0.660263$
Root an. cond. $0.812565$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)4-s − 1.73i·13-s + (−0.499 − 0.866i)16-s + (−0.5 + 0.866i)25-s + (1.5 + 0.866i)31-s + (−0.5 − 0.866i)37-s + 43-s + (−1.49 − 0.866i)52-s + (−1.5 + 0.866i)61-s − 0.999·64-s + (0.5 − 0.866i)67-s + (0.5 + 0.866i)79-s + 1.73i·97-s + (0.499 + 0.866i)100-s + (1.5 − 0.866i)103-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)4-s − 1.73i·13-s + (−0.499 − 0.866i)16-s + (−0.5 + 0.866i)25-s + (1.5 + 0.866i)31-s + (−0.5 − 0.866i)37-s + 43-s + (−1.49 − 0.866i)52-s + (−1.5 + 0.866i)61-s − 0.999·64-s + (0.5 − 0.866i)67-s + (0.5 + 0.866i)79-s + 1.73i·97-s + (0.499 + 0.866i)100-s + (1.5 − 0.866i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.444 + 0.895i$
Analytic conductor: \(0.660263\)
Root analytic conductor: \(0.812565\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :0),\ 0.444 + 0.895i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.174148301\)
\(L(\frac12)\) \(\approx\) \(1.174148301\)
\(L(1)\) 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 \( 1 + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + 1.73iT - T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 - 1.73iT - T^{2} \)
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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

−9.843553237222732707474108860594, −8.974816365916546114042264734303, −7.953123816083389962894659645922, −7.26190323170567542077391119980, −6.24333928291669579276411309398, −5.57645746564939591781772609965, −4.82221048394423545007869832824, −3.41928177536144544564167779469, −2.44011166590495548838277642670, −1.05989637758594156347961776512, 1.85444917602858520700677709570, 2.81924541317498148269628843430, 4.00715638032429347033310768449, 4.61935737178106725755288828837, 6.13596508737825238239272358710, 6.67264872318306544370824332564, 7.55243301252859911976984732658, 8.309353841976895473163553570025, 9.075637620829755838827725360021, 9.908404490530258951916925184214

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