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Properties

Label 2-1323-63.59-c1-0-25
Degree $2$
Conductor $1323$
Sign $0.460 + 0.887i$
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  + (0.575 + 0.332i)2-s + (−0.779 − 1.34i)4-s − 0.0283·5-s − 2.36i·8-s + (−0.0162 − 0.00940i)10-s + 1.02i·11-s + (4.87 + 2.81i)13-s + (−0.773 + 1.33i)16-s + (2.83 − 4.91i)17-s + (−1.81 + 1.04i)19-s + (0.0220 + 0.0382i)20-s + (−0.339 + 0.588i)22-s − 7.26i·23-s − 4.99·25-s + (1.87 + 3.24i)26-s + ⋯
L(s)  = 1  + (0.406 + 0.234i)2-s + (−0.389 − 0.674i)4-s − 0.0126·5-s − 0.835i·8-s + (−0.00514 − 0.00297i)10-s + 0.308i·11-s + (1.35 + 0.781i)13-s + (−0.193 + 0.334i)16-s + (0.688 − 1.19i)17-s + (−0.415 + 0.240i)19-s + (0.00493 + 0.00854i)20-s + (−0.0723 + 0.125i)22-s − 1.51i·23-s − 0.999·25-s + (0.366 + 0.635i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.460 + 0.887i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.460 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.460 + 0.887i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (962, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ 0.460 + 0.887i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.789878692\)
\(L(\frac12)\) \(\approx\) \(1.789878692\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.575 - 0.332i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + 0.0283T + 5T^{2} \)
11 \( 1 - 1.02iT - 11T^{2} \)
13 \( 1 + (-4.87 - 2.81i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.83 + 4.91i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.81 - 1.04i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 7.26iT - 23T^{2} \)
29 \( 1 + (-3.52 + 2.03i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.87 + 1.65i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.23 - 2.14i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.52 + 6.11i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.15 + 2.00i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.43 + 9.42i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (10.0 + 5.79i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.01 + 5.21i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.05 - 1.18i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.38 + 11.0i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.93iT - 71T^{2} \)
73 \( 1 + (-9.43 - 5.44i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.80 + 13.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.07 - 5.32i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-6.02 - 10.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.77 - 3.91i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.505006767478356107550363190389, −8.767913125034207507137438702663, −7.88804431868005154134011111518, −6.68204864545784950527476118883, −6.25451387091144157801386314436, −5.25227337544451476617892893254, −4.41739267484002771389147962362, −3.63897656136666419557690761480, −2.12040883710923579122481684192, −0.72930577790826048841837599794, 1.38373738457121131480170305329, 2.96568731810466242360867347149, 3.63118479983564631210671163696, 4.46608076505577240818688091072, 5.66042018727546608098202951522, 6.18060364248999578489881364903, 7.65191080221246146257680060318, 8.093182275220626371897340421994, 8.841421524817106479695458262614, 9.733461441489594578869222506820

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