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Properties

Label 2-33-3.2-c6-0-7
Degree $2$
Conductor $33$
Sign $0.622 - 0.782i$
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  + 8.58i·2-s + (−21.1 − 16.8i)3-s − 9.71·4-s − 61.7i·5-s + (144. − 181. i)6-s + 427.·7-s + 466. i·8-s + (163. + 710. i)9-s + 530.·10-s − 401. i·11-s + (205. + 163. i)12-s + 2.36e3·13-s + 3.67e3i·14-s + (−1.03e3 + 1.30e3i)15-s − 4.62e3·16-s + 4.07e3i·17-s + ⋯
L(s)  = 1  + 1.07i·2-s + (−0.782 − 0.622i)3-s − 0.151·4-s − 0.494i·5-s + (0.668 − 0.839i)6-s + 1.24·7-s + 0.910i·8-s + (0.224 + 0.974i)9-s + 0.530·10-s − 0.301i·11-s + (0.118 + 0.0945i)12-s + 1.07·13-s + 1.33i·14-s + (−0.307 + 0.386i)15-s − 1.12·16-s + 0.828i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.622 - 0.782i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.622 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $0.622 - 0.782i$
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: \((2,\ 33,\ (\ :3),\ 0.622 - 0.782i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.47936 + 0.713313i\)
\(L(\frac12)\) \(\approx\) \(1.47936 + 0.713313i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (21.1 + 16.8i)T \)
11 \( 1 + 401. iT \)
good2 \( 1 - 8.58iT - 64T^{2} \)
5 \( 1 + 61.7iT - 1.56e4T^{2} \)
7 \( 1 - 427.T + 1.17e5T^{2} \)
13 \( 1 - 2.36e3T + 4.82e6T^{2} \)
17 \( 1 - 4.07e3iT - 2.41e7T^{2} \)
19 \( 1 - 1.26e4T + 4.70e7T^{2} \)
23 \( 1 + 4.64e3iT - 1.48e8T^{2} \)
29 \( 1 + 2.60e4iT - 5.94e8T^{2} \)
31 \( 1 + 3.32e4T + 8.87e8T^{2} \)
37 \( 1 + 2.47e4T + 2.56e9T^{2} \)
41 \( 1 - 6.35e4iT - 4.75e9T^{2} \)
43 \( 1 + 2.75e4T + 6.32e9T^{2} \)
47 \( 1 - 6.74e4iT - 1.07e10T^{2} \)
53 \( 1 + 1.67e5iT - 2.21e10T^{2} \)
59 \( 1 - 2.25e5iT - 4.21e10T^{2} \)
61 \( 1 + 2.89e5T + 5.15e10T^{2} \)
67 \( 1 - 2.35e5T + 9.04e10T^{2} \)
71 \( 1 - 1.53e5iT - 1.28e11T^{2} \)
73 \( 1 + 3.06e5T + 1.51e11T^{2} \)
79 \( 1 - 6.37e5T + 2.43e11T^{2} \)
83 \( 1 + 9.34e5iT - 3.26e11T^{2} \)
89 \( 1 + 9.53e5iT - 4.96e11T^{2} \)
97 \( 1 + 7.23e5T + 8.32e11T^{2} \)
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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

−15.89567920858130228358602382416, −14.49919016019891624646067311790, −13.37563614655277530916847625137, −11.78282810655568752686046507123, −10.94866568224473509236355020072, −8.487537599937041178061259467097, −7.56315573614484156108040603987, −6.06120214397264579389273216433, −5.00792378864423345818898699504, −1.40284129263028339754097468489, 1.26655076536953695064935164733, 3.45773451858614783048695349426, 5.17902344917777374437148709320, 7.10421377273396593068913682789, 9.320723342095004646692370132080, 10.71355460164612743299675794142, 11.27500194691282969879763689227, 12.21740034543655640851171295414, 13.97271463946403146247800414990, 15.37313729880304063688238365465

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