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Properties

Label 2-1323-21.20-c1-0-46
Degree $2$
Conductor $1323$
Sign $-0.987 - 0.156i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.70i·2-s − 0.911·4-s − 0.829·5-s − 1.85i·8-s + 1.41i·10-s − 0.190i·11-s − 3.53i·13-s − 4.99·16-s + 4.60·17-s + 4.09i·19-s + 0.755·20-s − 0.325·22-s − 8.79i·23-s − 4.31·25-s − 6.02·26-s + ⋯
L(s)  = 1  − 1.20i·2-s − 0.455·4-s − 0.370·5-s − 0.656i·8-s + 0.447i·10-s − 0.0575i·11-s − 0.979i·13-s − 1.24·16-s + 1.11·17-s + 0.939i·19-s + 0.169·20-s − 0.0694·22-s − 1.83i·23-s − 0.862·25-s − 1.18·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.156i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 - 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.987 - 0.156i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (1322, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ -0.987 - 0.156i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.230835687\)
\(L(\frac12)\) \(\approx\) \(1.230835687\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.70iT - 2T^{2} \)
5 \( 1 + 0.829T + 5T^{2} \)
11 \( 1 + 0.190iT - 11T^{2} \)
13 \( 1 + 3.53iT - 13T^{2} \)
17 \( 1 - 4.60T + 17T^{2} \)
19 \( 1 - 4.09iT - 19T^{2} \)
23 \( 1 + 8.79iT - 23T^{2} \)
29 \( 1 + 7.99iT - 29T^{2} \)
31 \( 1 - 6.56iT - 31T^{2} \)
37 \( 1 - 1.45T + 37T^{2} \)
41 \( 1 + 12.1T + 41T^{2} \)
43 \( 1 - 2.83T + 43T^{2} \)
47 \( 1 + 1.49T + 47T^{2} \)
53 \( 1 + 4.55iT - 53T^{2} \)
59 \( 1 + 5.67T + 59T^{2} \)
61 \( 1 + 10.9iT - 61T^{2} \)
67 \( 1 + 4.64T + 67T^{2} \)
71 \( 1 + 10.0iT - 71T^{2} \)
73 \( 1 - 7.95iT - 73T^{2} \)
79 \( 1 + 5.96T + 79T^{2} \)
83 \( 1 - 2.05T + 83T^{2} \)
89 \( 1 - 8.83T + 89T^{2} \)
97 \( 1 + 7.15iT - 97T^{2} \)
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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

−9.582111149685944264644825844918, −8.360464632602239381598647073363, −7.84808468550901003446577960483, −6.71134035510808598334335896269, −5.81341839636860255214231253097, −4.66153300540860193221606678242, −3.65840290820617382444321865895, −2.98696145245813336097628015591, −1.81744344042734391403438379278, −0.49986314725576766955275373858, 1.70408906827972439582090656204, 3.16792459190941378371132266091, 4.29826502563101278617539237900, 5.28721332433857818037718726371, 5.94197659052400697390953091595, 7.00943710863674542025920282876, 7.40668223578777599584516971393, 8.208466465837216370886184399662, 9.088580811127193758136004910314, 9.737730811586443228130337664160

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