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Properties

Label 2-33-3.2-c8-0-13
Degree $2$
Conductor $33$
Sign $0.316 + 0.948i$
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  − 22.5i·2-s + (76.8 − 25.6i)3-s − 252.·4-s + 1.14e3i·5-s + (−577. − 1.73e3i)6-s + 3.91e3·7-s − 73.9i·8-s + (5.24e3 − 3.93e3i)9-s + 2.59e4·10-s + 4.41e3i·11-s + (−1.94e4 + 6.47e3i)12-s + 1.07e4·13-s − 8.83e4i·14-s + (2.94e4 + 8.83e4i)15-s − 6.63e4·16-s + 5.89e4i·17-s + ⋯
L(s)  = 1  − 1.40i·2-s + (0.948 − 0.316i)3-s − 0.987·4-s + 1.83i·5-s + (−0.445 − 1.33i)6-s + 1.63·7-s − 0.0180i·8-s + (0.799 − 0.600i)9-s + 2.59·10-s + 0.301i·11-s + (−0.936 + 0.312i)12-s + 0.377·13-s − 2.29i·14-s + (0.581 + 1.74i)15-s − 1.01·16-s + 0.706i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.316 + 0.948i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $0.316 + 0.948i$
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: \((2,\ 33,\ (\ :4),\ 0.316 + 0.948i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(2.36100 - 1.70156i\)
\(L(\frac12)\) \(\approx\) \(2.36100 - 1.70156i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-76.8 + 25.6i)T \)
11 \( 1 - 4.41e3iT \)
good2 \( 1 + 22.5iT - 256T^{2} \)
5 \( 1 - 1.14e3iT - 3.90e5T^{2} \)
7 \( 1 - 3.91e3T + 5.76e6T^{2} \)
13 \( 1 - 1.07e4T + 8.15e8T^{2} \)
17 \( 1 - 5.89e4iT - 6.97e9T^{2} \)
19 \( 1 - 4.52e4T + 1.69e10T^{2} \)
23 \( 1 + 2.47e5iT - 7.83e10T^{2} \)
29 \( 1 + 1.32e5iT - 5.00e11T^{2} \)
31 \( 1 - 2.28e5T + 8.52e11T^{2} \)
37 \( 1 - 9.43e5T + 3.51e12T^{2} \)
41 \( 1 + 1.19e6iT - 7.98e12T^{2} \)
43 \( 1 + 5.64e6T + 1.16e13T^{2} \)
47 \( 1 + 2.93e6iT - 2.38e13T^{2} \)
53 \( 1 + 7.26e6iT - 6.22e13T^{2} \)
59 \( 1 - 1.19e7iT - 1.46e14T^{2} \)
61 \( 1 - 8.48e6T + 1.91e14T^{2} \)
67 \( 1 + 3.04e7T + 4.06e14T^{2} \)
71 \( 1 - 2.17e7iT - 6.45e14T^{2} \)
73 \( 1 - 4.63e7T + 8.06e14T^{2} \)
79 \( 1 + 6.98e7T + 1.51e15T^{2} \)
83 \( 1 - 8.83e6iT - 2.25e15T^{2} \)
89 \( 1 - 7.69e7iT - 3.93e15T^{2} \)
97 \( 1 + 7.58e7T + 7.83e15T^{2} \)
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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

−14.50526304766802291459177788212, −13.51552621159598729262092575887, −11.89420116377646474773872977160, −10.93131179003187527858073951080, −10.04939121325682295272503206283, −8.256189347950738009409748887902, −6.90070294316849685052753205859, −3.97146445510183142920983255296, −2.65079764510053375310076805205, −1.65055827155671419181287289645, 1.46677245629984560233080582215, 4.54045675711153890640694074205, 5.31009353246687612447292094356, 7.73203963358341201933597171650, 8.369995433133341812006841899013, 9.253935124273410884682411952282, 11.56787614604297394612502331220, 13.35513352059636691819386103633, 14.15710724493518091571683344989, 15.33263460483417369283687478062

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