Properties

Label 2-33-3.2-c8-0-3
Degree $2$
Conductor $33$
Sign $-0.890 + 0.455i$
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  + 14.0i·2-s + (36.8 + 72.1i)3-s + 59.4·4-s − 519. i·5-s + (−1.01e3 + 516. i)6-s − 3.94e3·7-s + 4.42e3i·8-s + (−3.84e3 + 5.31e3i)9-s + 7.28e3·10-s + 4.41e3i·11-s + (2.19e3 + 4.28e3i)12-s − 4.12e4·13-s − 5.53e4i·14-s + (3.74e4 − 1.91e4i)15-s − 4.67e4·16-s + 5.58e4i·17-s + ⋯
L(s)  = 1  + 0.876i·2-s + (0.455 + 0.890i)3-s + 0.232·4-s − 0.831i·5-s + (−0.780 + 0.398i)6-s − 1.64·7-s + 1.07i·8-s + (−0.585 + 0.810i)9-s + 0.728·10-s + 0.301i·11-s + (0.105 + 0.206i)12-s − 1.44·13-s − 1.44i·14-s + (0.740 − 0.378i)15-s − 0.713·16-s + 0.668i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.890 + 0.455i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.890 + 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $-0.890 + 0.455i$
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: \((2,\ 33,\ (\ :4),\ -0.890 + 0.455i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.260873 - 1.08330i\)
\(L(\frac12)\) \(\approx\) \(0.260873 - 1.08330i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-36.8 - 72.1i)T \)
11 \( 1 - 4.41e3iT \)
good2 \( 1 - 14.0iT - 256T^{2} \)
5 \( 1 + 519. iT - 3.90e5T^{2} \)
7 \( 1 + 3.94e3T + 5.76e6T^{2} \)
13 \( 1 + 4.12e4T + 8.15e8T^{2} \)
17 \( 1 - 5.58e4iT - 6.97e9T^{2} \)
19 \( 1 - 3.93e3T + 1.69e10T^{2} \)
23 \( 1 - 1.52e5iT - 7.83e10T^{2} \)
29 \( 1 + 6.84e5iT - 5.00e11T^{2} \)
31 \( 1 - 1.38e6T + 8.52e11T^{2} \)
37 \( 1 - 5.64e5T + 3.51e12T^{2} \)
41 \( 1 - 4.96e6iT - 7.98e12T^{2} \)
43 \( 1 + 1.20e4T + 1.16e13T^{2} \)
47 \( 1 + 7.11e5iT - 2.38e13T^{2} \)
53 \( 1 - 8.30e6iT - 6.22e13T^{2} \)
59 \( 1 - 2.73e6iT - 1.46e14T^{2} \)
61 \( 1 + 1.60e7T + 1.91e14T^{2} \)
67 \( 1 + 3.77e7T + 4.06e14T^{2} \)
71 \( 1 - 8.44e6iT - 6.45e14T^{2} \)
73 \( 1 - 1.44e7T + 8.06e14T^{2} \)
79 \( 1 - 6.67e7T + 1.51e15T^{2} \)
83 \( 1 + 6.89e7iT - 2.25e15T^{2} \)
89 \( 1 - 3.31e7iT - 3.93e15T^{2} \)
97 \( 1 + 6.75e7T + 7.83e15T^{2} \)
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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

−15.65509181587109521971651575945, −14.89269313673457498272952030058, −13.40585567998149690812164920338, −12.07650661183068414854635502206, −10.16181994693213977236933774523, −9.208497699470302990813661323599, −7.77650189786040332513141682568, −6.18926378582019875340707879595, −4.69759705151898388419037338089, −2.76706770251155172737910169731, 0.40067130807906440954140747680, 2.52353844197027997064691385343, 3.20314210319258847467486297427, 6.49060419428749591113929734210, 7.20075714785994519884373536674, 9.376190365618837128953060092857, 10.44321792792820899763575553480, 11.99666207124706211901065371297, 12.71490692111996208412550080853, 13.93957496577295524519851740243

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