Properties

Label 2-33-3.2-c6-0-15
Degree $2$
Conductor $33$
Sign $0.518 + 0.855i$
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  + 2.50i·2-s + (23.0 − 13.9i)3-s + 57.7·4-s − 201. i·5-s + (35.0 + 57.8i)6-s − 481.·7-s + 305. i·8-s + (337. − 646. i)9-s + 504.·10-s − 401. i·11-s + (1.33e3 − 807. i)12-s + 552.·13-s − 1.20e3i·14-s + (−2.81e3 − 4.64e3i)15-s + 2.92e3·16-s + 2.69e3i·17-s + ⋯
L(s)  = 1  + 0.313i·2-s + (0.855 − 0.518i)3-s + 0.901·4-s − 1.61i·5-s + (0.162 + 0.267i)6-s − 1.40·7-s + 0.595i·8-s + (0.462 − 0.886i)9-s + 0.504·10-s − 0.301i·11-s + (0.771 − 0.467i)12-s + 0.251·13-s − 0.440i·14-s + (−0.835 − 1.37i)15-s + 0.715·16-s + 0.549i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.518 + 0.855i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.518 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $0.518 + 0.855i$
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: \((2,\ 33,\ (\ :3),\ 0.518 + 0.855i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(2.02837 - 1.14222i\)
\(L(\frac12)\) \(\approx\) \(2.02837 - 1.14222i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-23.0 + 13.9i)T \)
11 \( 1 + 401. iT \)
good2 \( 1 - 2.50iT - 64T^{2} \)
5 \( 1 + 201. iT - 1.56e4T^{2} \)
7 \( 1 + 481.T + 1.17e5T^{2} \)
13 \( 1 - 552.T + 4.82e6T^{2} \)
17 \( 1 - 2.69e3iT - 2.41e7T^{2} \)
19 \( 1 - 1.21e4T + 4.70e7T^{2} \)
23 \( 1 - 1.24e4iT - 1.48e8T^{2} \)
29 \( 1 - 2.01e4iT - 5.94e8T^{2} \)
31 \( 1 - 2.07e4T + 8.87e8T^{2} \)
37 \( 1 + 1.55e4T + 2.56e9T^{2} \)
41 \( 1 + 5.95e4iT - 4.75e9T^{2} \)
43 \( 1 + 2.50e4T + 6.32e9T^{2} \)
47 \( 1 + 2.24e4iT - 1.07e10T^{2} \)
53 \( 1 - 2.76e5iT - 2.21e10T^{2} \)
59 \( 1 + 2.45e5iT - 4.21e10T^{2} \)
61 \( 1 + 2.32e4T + 5.15e10T^{2} \)
67 \( 1 - 6.81e4T + 9.04e10T^{2} \)
71 \( 1 - 5.72e5iT - 1.28e11T^{2} \)
73 \( 1 + 5.21e5T + 1.51e11T^{2} \)
79 \( 1 + 4.68e5T + 2.43e11T^{2} \)
83 \( 1 - 9.27e4iT - 3.26e11T^{2} \)
89 \( 1 + 4.22e5iT - 4.96e11T^{2} \)
97 \( 1 - 1.11e6T + 8.32e11T^{2} \)
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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

−15.65952285624295728358574093192, −13.83523776283496448634116693390, −12.84044711380304892259642077125, −11.95693626245109635875490425280, −9.724235812177245850146432601310, −8.619797525683427433315474636664, −7.28673272088362735026357513588, −5.78817678518565028700313931134, −3.30308367928658891970159127048, −1.24257558289054824112599500717, 2.66354252057629567950270494601, 3.37066066108152151247637972052, 6.46064139315193212947631721948, 7.45705006631421755202117901652, 9.722076192042119495018829626671, 10.34953355560630499631030864523, 11.68518370160297657299582487607, 13.37531005205724825685978980174, 14.57220522076463718274102596649, 15.60723569236630603528313067980

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