Properties

Label 2-1323-147.137-c0-0-0
Degree $2$
Conductor $1323$
Sign $0.991 + 0.127i$
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  + (0.955 + 0.294i)4-s + (0.623 − 0.781i)7-s + (0.0931 − 0.116i)13-s + (0.826 + 0.563i)16-s + (−0.623 − 1.07i)19-s + (−0.988 + 0.149i)25-s + (0.826 − 0.563i)28-s + (−0.955 + 1.65i)31-s + (1.82 − 0.563i)37-s + (−0.658 + 0.317i)43-s + (−0.222 − 0.974i)49-s + (0.123 − 0.0841i)52-s + (−1.40 + 0.432i)61-s + (0.623 + 0.781i)64-s + (−0.826 + 1.43i)67-s + ⋯
L(s)  = 1  + (0.955 + 0.294i)4-s + (0.623 − 0.781i)7-s + (0.0931 − 0.116i)13-s + (0.826 + 0.563i)16-s + (−0.623 − 1.07i)19-s + (−0.988 + 0.149i)25-s + (0.826 − 0.563i)28-s + (−0.955 + 1.65i)31-s + (1.82 − 0.563i)37-s + (−0.658 + 0.317i)43-s + (−0.222 − 0.974i)49-s + (0.123 − 0.0841i)52-s + (−1.40 + 0.432i)61-s + (0.623 + 0.781i)64-s + (−0.826 + 1.43i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.991 + 0.127i$
Analytic conductor: \(0.660263\)
Root analytic conductor: \(0.812565\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :0),\ 0.991 + 0.127i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.401849417\)
\(L(\frac12)\) \(\approx\) \(1.401849417\)
\(L(1)\) 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 \( 1 + (-0.623 + 0.781i)T \)
good2 \( 1 + (-0.955 - 0.294i)T^{2} \)
5 \( 1 + (0.988 - 0.149i)T^{2} \)
11 \( 1 + (0.733 - 0.680i)T^{2} \)
13 \( 1 + (-0.0931 + 0.116i)T + (-0.222 - 0.974i)T^{2} \)
17 \( 1 + (-0.0747 - 0.997i)T^{2} \)
19 \( 1 + (0.623 + 1.07i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.0747 + 0.997i)T^{2} \)
29 \( 1 + (0.900 + 0.433i)T^{2} \)
31 \( 1 + (0.955 - 1.65i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-1.82 + 0.563i)T + (0.826 - 0.563i)T^{2} \)
41 \( 1 + (-0.623 - 0.781i)T^{2} \)
43 \( 1 + (0.658 - 0.317i)T + (0.623 - 0.781i)T^{2} \)
47 \( 1 + (-0.955 - 0.294i)T^{2} \)
53 \( 1 + (-0.826 - 0.563i)T^{2} \)
59 \( 1 + (0.988 + 0.149i)T^{2} \)
61 \( 1 + (1.40 - 0.432i)T + (0.826 - 0.563i)T^{2} \)
67 \( 1 + (0.826 - 1.43i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.900 - 0.433i)T^{2} \)
73 \( 1 + (-0.440 + 0.0663i)T + (0.955 - 0.294i)T^{2} \)
79 \( 1 + (-0.733 - 1.26i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.222 - 0.974i)T^{2} \)
89 \( 1 + (0.733 + 0.680i)T^{2} \)
97 \( 1 - 0.730T + T^{2} \)
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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

−9.964484187881104830187027019787, −8.898858729439210072619014515637, −8.025636077429588630974360727520, −7.35432147502833390700002261100, −6.69972499027653734875214174790, −5.74476523499981902188562181099, −4.64453880334471008996235610463, −3.70429982610737716866988015480, −2.60375470571009684141867570961, −1.46327776776313229163270577211, 1.69263510353415156345136098572, 2.43189292073389740998427817482, 3.71600699375093988717230750029, 4.88027351676927085870128965539, 6.00562923699590937731487376015, 6.20525788248326433027249029302, 7.66596976799529336451474911651, 7.940643047489370963387158194880, 9.107732001437310395481740746971, 9.870968251957206685687320060444

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