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Properties

Label 2-1323-49.5-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 + (−1.55 + 1.24i)13-s + (0.826 − 0.563i)16-s + (−1.35 − 0.781i)19-s + (−0.988 − 0.149i)25-s + (0.826 + 0.563i)28-s + (−0.510 + 0.294i)31-s + (−1.82 − 0.563i)37-s + (0.658 + 0.317i)43-s + (−0.222 + 0.974i)49-s + (1.12 − 1.64i)52-s + (−0.400 + 1.29i)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 + (−1.55 + 1.24i)13-s + (0.826 − 0.563i)16-s + (−1.35 − 0.781i)19-s + (−0.988 − 0.149i)25-s + (0.826 + 0.563i)28-s + (−0.510 + 0.294i)31-s + (−1.82 − 0.563i)37-s + (0.658 + 0.317i)43-s + (−0.222 + 0.974i)49-s + (1.12 − 1.64i)52-s + (−0.400 + 1.29i)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: $\chi_{1323} (838, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :0),\ -0.991 + 0.127i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03662704457\)
\(L(\frac12)\) \(\approx\) \(0.03662704457\)
\(L(1)\) 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 \( 1 + (1.55 - 1.24i)T + (0.222 - 0.974i)T^{2} \)
17 \( 1 + (-0.0747 + 0.997i)T^{2} \)
19 \( 1 + (1.35 + 0.781i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.0747 + 0.997i)T^{2} \)
29 \( 1 + (-0.900 + 0.433i)T^{2} \)
31 \( 1 + (0.510 - 0.294i)T + (0.5 - 0.866i)T^{2} \)
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 \( 1 + (0.400 - 1.29i)T + (-0.826 - 0.563i)T^{2} \)
67 \( 1 + (0.826 + 1.43i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.900 - 0.433i)T^{2} \)
73 \( 1 + (-0.290 + 1.92i)T + (-0.955 - 0.294i)T^{2} \)
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 \( 1 - 1.86iT - T^{2} \)
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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

−9.285202537459896907585351816670, −8.928516194056979783755897098001, −7.69601541956792575137029961892, −7.12326876490299121492575828079, −6.21903540329620238400613616360, −4.90906333092754311942445387534, −4.34983273584192560438597950498, −3.47366781428030281148884648419, −2.12527774224179111093683895933, −0.02943870749834837530785946662, 2.08501646360362318764886522573, 3.26294381228667612873956131774, 4.29067912492607441260120878451, 5.37336825809673311079390856628, 5.76840182699086055063492173057, 6.93612451540163817440942064275, 8.015574611877370563693035438516, 8.597121852866722960107024061396, 9.550154350493744470019830217791, 10.00967600555923561793443100810

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