Properties

Label 2-33-11.10-c6-0-7
Degree $2$
Conductor $33$
Sign $0.568 + 0.822i$
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  + 14.5i·2-s − 15.5·3-s − 148.·4-s + 0.687·5-s − 227. i·6-s − 18.0i·7-s − 1.23e3i·8-s + 243·9-s + 10.0i·10-s + (−756. − 1.09e3i)11-s + 2.32e3·12-s + 2.82e3i·13-s + 262.·14-s − 10.7·15-s + 8.55e3·16-s − 2.97e3i·17-s + ⋯
L(s)  = 1  + 1.82i·2-s − 0.577·3-s − 2.32·4-s + 0.00550·5-s − 1.05i·6-s − 0.0525i·7-s − 2.42i·8-s + 0.333·9-s + 0.0100i·10-s + (−0.568 − 0.822i)11-s + 1.34·12-s + 1.28i·13-s + 0.0957·14-s − 0.00317·15-s + 2.08·16-s − 0.605i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.568 + 0.822i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.568 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $0.568 + 0.822i$
Analytic conductor: \(7.59178\)
Root analytic conductor: \(2.75531\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :3),\ 0.568 + 0.822i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.000749452 - 0.000393182i\)
\(L(\frac12)\) \(\approx\) \(0.000749452 - 0.000393182i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 15.5T \)
11 \( 1 + (756. + 1.09e3i)T \)
good2 \( 1 - 14.5iT - 64T^{2} \)
5 \( 1 - 0.687T + 1.56e4T^{2} \)
7 \( 1 + 18.0iT - 1.17e5T^{2} \)
13 \( 1 - 2.82e3iT - 4.82e6T^{2} \)
17 \( 1 + 2.97e3iT - 2.41e7T^{2} \)
19 \( 1 + 8.40e3iT - 4.70e7T^{2} \)
23 \( 1 + 1.97e4T + 1.48e8T^{2} \)
29 \( 1 + 8.49e3iT - 5.94e8T^{2} \)
31 \( 1 - 2.26e4T + 8.87e8T^{2} \)
37 \( 1 + 8.83e4T + 2.56e9T^{2} \)
41 \( 1 - 3.41e4iT - 4.75e9T^{2} \)
43 \( 1 - 1.31e5iT - 6.32e9T^{2} \)
47 \( 1 + 7.55e4T + 1.07e10T^{2} \)
53 \( 1 + 1.53e5T + 2.21e10T^{2} \)
59 \( 1 - 2.03e5T + 4.21e10T^{2} \)
61 \( 1 - 1.41e5iT - 5.15e10T^{2} \)
67 \( 1 - 1.63e5T + 9.04e10T^{2} \)
71 \( 1 + 3.04e5T + 1.28e11T^{2} \)
73 \( 1 - 5.97e5iT - 1.51e11T^{2} \)
79 \( 1 + 2.22e5iT - 2.43e11T^{2} \)
83 \( 1 + 7.18e5iT - 3.26e11T^{2} \)
89 \( 1 + 1.24e6T + 4.96e11T^{2} \)
97 \( 1 + 1.43e5T + 8.32e11T^{2} \)
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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

−15.72771226203447015029561428538, −14.15583594534646889812125596934, −13.41525904248760842225405369574, −11.59847183646276988873366365997, −9.691798073277909240586560590191, −8.320861504384774973210894329732, −6.96783082834047170789225713804, −5.85739221170630965555202580293, −4.49072428382574826542811934492, −0.00047192508862913396893149720, 1.89791084913676605870523943980, 3.79639424741695901015245695918, 5.44734230132573887943548551328, 8.132072795981122699137162484179, 10.03039556925947035704789576998, 10.47744616120847609179437041437, 12.03742720556405985836736222290, 12.58365694074713861010791577326, 13.86765345191041964595666888313, 15.54572151025151120271105700514

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