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Properties

Label 2-33-3.2-c6-0-1
Degree $2$
Conductor $33$
Sign $-0.987 + 0.155i$
Analytic cond. $7.59178$
Root an. cond. $2.75531$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9.18i·2-s + (−4.20 − 26.6i)3-s − 20.2·4-s + 92.7i·5-s + (244. − 38.5i)6-s − 527.·7-s + 401. i·8-s + (−693. + 224. i)9-s − 851.·10-s + 401. i·11-s + (85.2 + 541. i)12-s − 1.31e3·13-s − 4.84e3i·14-s + (2.47e3 − 389. i)15-s − 4.98e3·16-s − 3.88e3i·17-s + ⋯
L(s)  = 1  + 1.14i·2-s + (−0.155 − 0.987i)3-s − 0.316·4-s + 0.741i·5-s + (1.13 − 0.178i)6-s − 1.53·7-s + 0.783i·8-s + (−0.951 + 0.307i)9-s − 0.851·10-s + 0.301i·11-s + (0.0493 + 0.313i)12-s − 0.600·13-s − 1.76i·14-s + (0.732 − 0.115i)15-s − 1.21·16-s − 0.790i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.155i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.987 + 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $-0.987 + 0.155i$
Analytic conductor: \(7.59178\)
Root analytic conductor: \(2.75531\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :3),\ -0.987 + 0.155i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.0472754 - 0.604036i\)
\(L(\frac12)\) \(\approx\) \(0.0472754 - 0.604036i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.20 + 26.6i)T \)
11 \( 1 - 401. iT \)
good2 \( 1 - 9.18iT - 64T^{2} \)
5 \( 1 - 92.7iT - 1.56e4T^{2} \)
7 \( 1 + 527.T + 1.17e5T^{2} \)
13 \( 1 + 1.31e3T + 4.82e6T^{2} \)
17 \( 1 + 3.88e3iT - 2.41e7T^{2} \)
19 \( 1 + 2.93e3T + 4.70e7T^{2} \)
23 \( 1 - 2.09e4iT - 1.48e8T^{2} \)
29 \( 1 + 4.50e3iT - 5.94e8T^{2} \)
31 \( 1 + 2.97e4T + 8.87e8T^{2} \)
37 \( 1 - 9.63e4T + 2.56e9T^{2} \)
41 \( 1 + 6.50e4iT - 4.75e9T^{2} \)
43 \( 1 - 2.63e4T + 6.32e9T^{2} \)
47 \( 1 - 1.21e5iT - 1.07e10T^{2} \)
53 \( 1 + 7.77e4iT - 2.21e10T^{2} \)
59 \( 1 - 2.79e4iT - 4.21e10T^{2} \)
61 \( 1 + 3.81e5T + 5.15e10T^{2} \)
67 \( 1 - 2.83e5T + 9.04e10T^{2} \)
71 \( 1 - 3.02e5iT - 1.28e11T^{2} \)
73 \( 1 + 2.91e5T + 1.51e11T^{2} \)
79 \( 1 + 4.63e5T + 2.43e11T^{2} \)
83 \( 1 - 8.69e5iT - 3.26e11T^{2} \)
89 \( 1 - 1.62e5iT - 4.96e11T^{2} \)
97 \( 1 + 1.08e6T + 8.32e11T^{2} \)
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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

−16.09671081727824799274643940136, −14.93948882243653925498673440339, −13.81598343558251048318714214931, −12.65691221283981327891143680383, −11.23578905029345236974677993224, −9.412814579949494866343222088456, −7.52500129292268851566709095007, −6.83613240162881611535879578827, −5.76709683798955960314375602251, −2.71904751980583326555788449079, 0.29948894578299191284007519485, 2.89737365154838754923590688213, 4.28859114428117122945598599871, 6.28475110042581358970699474356, 8.928397590524989644929831628720, 9.906452980298958019878645843327, 10.83735759222071585502818817490, 12.33063767073650036875208098274, 13.00263191979625219182254340366, 14.92663140049110164908770441733

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