Properties

Label 2-33-3.2-c8-0-9
Degree $2$
Conductor $33$
Sign $0.531 - 0.847i$
Analytic cond. $13.4434$
Root an. cond. $3.66653$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 27.8i·2-s + (−68.6 − 43.0i)3-s − 519.·4-s − 436. i·5-s + (1.19e3 − 1.91e3i)6-s − 547.·7-s − 7.33e3i·8-s + (2.85e3 + 5.90e3i)9-s + 1.21e4·10-s + 4.41e3i·11-s + (3.56e4 + 2.23e4i)12-s + 2.44e4·13-s − 1.52e4i·14-s + (−1.88e4 + 2.99e4i)15-s + 7.13e4·16-s − 1.03e5i·17-s + ⋯
L(s)  = 1  + 1.74i·2-s + (−0.847 − 0.531i)3-s − 2.02·4-s − 0.699i·5-s + (0.924 − 1.47i)6-s − 0.227·7-s − 1.79i·8-s + (0.435 + 0.900i)9-s + 1.21·10-s + 0.301i·11-s + (1.71 + 1.07i)12-s + 0.856·13-s − 0.396i·14-s + (−0.371 + 0.592i)15-s + 1.08·16-s − 1.23i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.531 - 0.847i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.531 - 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $0.531 - 0.847i$
Analytic conductor: \(13.4434\)
Root analytic conductor: \(3.66653\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :4),\ 0.531 - 0.847i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.952474 + 0.526966i\)
\(L(\frac12)\) \(\approx\) \(0.952474 + 0.526966i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (68.6 + 43.0i)T \)
11 \( 1 - 4.41e3iT \)
good2 \( 1 - 27.8iT - 256T^{2} \)
5 \( 1 + 436. iT - 3.90e5T^{2} \)
7 \( 1 + 547.T + 5.76e6T^{2} \)
13 \( 1 - 2.44e4T + 8.15e8T^{2} \)
17 \( 1 + 1.03e5iT - 6.97e9T^{2} \)
19 \( 1 - 5.14e4T + 1.69e10T^{2} \)
23 \( 1 - 3.82e5iT - 7.83e10T^{2} \)
29 \( 1 + 6.63e5iT - 5.00e11T^{2} \)
31 \( 1 - 1.75e6T + 8.52e11T^{2} \)
37 \( 1 + 1.59e5T + 3.51e12T^{2} \)
41 \( 1 - 6.79e5iT - 7.98e12T^{2} \)
43 \( 1 - 6.20e6T + 1.16e13T^{2} \)
47 \( 1 + 3.90e6iT - 2.38e13T^{2} \)
53 \( 1 + 5.24e6iT - 6.22e13T^{2} \)
59 \( 1 + 1.59e7iT - 1.46e14T^{2} \)
61 \( 1 - 1.39e7T + 1.91e14T^{2} \)
67 \( 1 + 2.43e7T + 4.06e14T^{2} \)
71 \( 1 - 9.07e6iT - 6.45e14T^{2} \)
73 \( 1 + 2.81e7T + 8.06e14T^{2} \)
79 \( 1 - 1.11e7T + 1.51e15T^{2} \)
83 \( 1 - 6.75e7iT - 2.25e15T^{2} \)
89 \( 1 + 1.58e7iT - 3.93e15T^{2} \)
97 \( 1 - 1.31e8T + 7.83e15T^{2} \)
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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

−15.67518959012912375167537955777, −13.88727800124096444607472322079, −13.11466206522133923875717804099, −11.65355180254096543076708662228, −9.573095726647716593639543910276, −8.129782572365603885738714220899, −6.97481571470699803462033677550, −5.78839062509700604945985440050, −4.70182665385326380360354140262, −0.72948193472529989068869700128, 0.990854595119770962249864236253, 3.05334414907068327915621228582, 4.33983893905304284987008514865, 6.24613702221467070768800729299, 8.876807106565906172902743687359, 10.39172364353349526437738020323, 10.76790497443635958523666392928, 11.96308339747348682143531829754, 12.99058231243767838144526978581, 14.45897277394399673730013734632

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