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Properties

Label 2-1323-147.65-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  + (−0.733 + 0.680i)4-s + (0.623 − 0.781i)7-s + (1.03 − 1.29i)13-s + (0.0747 − 0.997i)16-s + (−0.623 + 1.07i)19-s + (0.365 − 0.930i)25-s + (0.0747 + 0.997i)28-s + (0.733 + 1.26i)31-s + (1.07 + 0.997i)37-s + (1.78 − 0.858i)43-s + (−0.222 − 0.974i)49-s + (0.123 + 1.64i)52-s + (−1.40 − 1.29i)61-s + (0.623 + 0.781i)64-s + (−0.0747 − 0.129i)67-s + ⋯
L(s)  = 1  + (−0.733 + 0.680i)4-s + (0.623 − 0.781i)7-s + (1.03 − 1.29i)13-s + (0.0747 − 0.997i)16-s + (−0.623 + 1.07i)19-s + (0.365 − 0.930i)25-s + (0.0747 + 0.997i)28-s + (0.733 + 1.26i)31-s + (1.07 + 0.997i)37-s + (1.78 − 0.858i)43-s + (−0.222 − 0.974i)49-s + (0.123 + 1.64i)52-s + (−1.40 − 1.29i)61-s + (0.623 + 0.781i)64-s + (−0.0747 − 0.129i)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: $\chi_{1323} (1241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :0),\ 0.991 + 0.127i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.006146023\)
\(L(\frac12)\) \(\approx\) \(1.006146023\)
\(L(1)\) 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 \( 1 + (0.733 - 0.680i)T^{2} \)
5 \( 1 + (-0.365 + 0.930i)T^{2} \)
11 \( 1 + (-0.955 - 0.294i)T^{2} \)
13 \( 1 + (-1.03 + 1.29i)T + (-0.222 - 0.974i)T^{2} \)
17 \( 1 + (-0.826 + 0.563i)T^{2} \)
19 \( 1 + (0.623 - 1.07i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.826 - 0.563i)T^{2} \)
29 \( 1 + (0.900 + 0.433i)T^{2} \)
31 \( 1 + (-0.733 - 1.26i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-1.07 - 0.997i)T + (0.0747 + 0.997i)T^{2} \)
41 \( 1 + (-0.623 - 0.781i)T^{2} \)
43 \( 1 + (-1.78 + 0.858i)T + (0.623 - 0.781i)T^{2} \)
47 \( 1 + (0.733 - 0.680i)T^{2} \)
53 \( 1 + (-0.0747 + 0.997i)T^{2} \)
59 \( 1 + (-0.365 - 0.930i)T^{2} \)
61 \( 1 + (1.40 + 1.29i)T + (0.0747 + 0.997i)T^{2} \)
67 \( 1 + (0.0747 + 0.129i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.900 - 0.433i)T^{2} \)
73 \( 1 + (0.162 - 0.414i)T + (-0.733 - 0.680i)T^{2} \)
79 \( 1 + (0.955 - 1.65i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.222 - 0.974i)T^{2} \)
89 \( 1 + (-0.955 + 0.294i)T^{2} \)
97 \( 1 + 1.97T + T^{2} \)
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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

−9.927443873192251474384247498063, −8.719374371964411806935556406155, −8.194673757154348924133802193781, −7.69382358362246232553965754371, −6.55264080167375170764091834919, −5.54528951127634503034720782977, −4.53770870968683742176111790247, −3.85389638871729008578180115535, −2.88228484127649967179953089190, −1.10073836243558224029702115256, 1.35156654757191696725858777478, 2.52503597256964043560335376931, 4.14998034349525144219043146844, 4.63209397528735178975803353349, 5.77819693799790605495662230015, 6.26215529781160173170747843915, 7.46810621730281730201333115941, 8.516232900563864339537288997118, 9.113012393487751304851119276855, 9.503322604358826957384342151569

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