Properties

Label 2-33-11.3-c7-0-6
Degree $2$
Conductor $33$
Sign $0.991 - 0.130i$
Analytic cond. $10.3087$
Root an. cond. $3.21071$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−12.1 + 8.81i)2-s + (8.34 + 25.6i)3-s + (29.9 − 92.0i)4-s + (−3.10 − 2.25i)5-s + (−327. − 237. i)6-s + (300. − 925. i)7-s + (−144. − 445. i)8-s + (−589. + 428. i)9-s + 57.5·10-s + (−3.92e3 − 2.01e3i)11-s + 2.61e3·12-s + (7.88e3 − 5.72e3i)13-s + (4.50e3 + 1.38e4i)14-s + (32.0 − 98.6i)15-s + (1.56e4 + 1.14e4i)16-s + (2.98e4 + 2.16e4i)17-s + ⋯
L(s)  = 1  + (−1.07 + 0.778i)2-s + (0.178 + 0.549i)3-s + (0.233 − 0.719i)4-s + (−0.0111 − 0.00807i)5-s + (−0.618 − 0.449i)6-s + (0.331 − 1.01i)7-s + (−0.0998 − 0.307i)8-s + (−0.269 + 0.195i)9-s + 0.0182·10-s + (−0.890 − 0.455i)11-s + 0.436·12-s + (0.994 − 0.722i)13-s + (0.438 + 1.35i)14-s + (0.00245 − 0.00754i)15-s + (0.958 + 0.696i)16-s + (1.47 + 1.06i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $0.991 - 0.130i$
Analytic conductor: \(10.3087\)
Root analytic conductor: \(3.21071\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :7/2),\ 0.991 - 0.130i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.918247 + 0.0603016i\)
\(L(\frac12)\) \(\approx\) \(0.918247 + 0.0603016i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-8.34 - 25.6i)T \)
11 \( 1 + (3.92e3 + 2.01e3i)T \)
good2 \( 1 + (12.1 - 8.81i)T + (39.5 - 121. i)T^{2} \)
5 \( 1 + (3.10 + 2.25i)T + (2.41e4 + 7.43e4i)T^{2} \)
7 \( 1 + (-300. + 925. i)T + (-6.66e5 - 4.84e5i)T^{2} \)
13 \( 1 + (-7.88e3 + 5.72e3i)T + (1.93e7 - 5.96e7i)T^{2} \)
17 \( 1 + (-2.98e4 - 2.16e4i)T + (1.26e8 + 3.90e8i)T^{2} \)
19 \( 1 + (3.26e3 + 1.00e4i)T + (-7.23e8 + 5.25e8i)T^{2} \)
23 \( 1 + 1.34e4T + 3.40e9T^{2} \)
29 \( 1 + (1.27e4 - 3.93e4i)T + (-1.39e10 - 1.01e10i)T^{2} \)
31 \( 1 + (-7.63e4 + 5.54e4i)T + (8.50e9 - 2.61e10i)T^{2} \)
37 \( 1 + (-1.45e5 + 4.47e5i)T + (-7.68e10 - 5.57e10i)T^{2} \)
41 \( 1 + (-1.36e4 - 4.19e4i)T + (-1.57e11 + 1.14e11i)T^{2} \)
43 \( 1 - 3.21e5T + 2.71e11T^{2} \)
47 \( 1 + (5.93e4 + 1.82e5i)T + (-4.09e11 + 2.97e11i)T^{2} \)
53 \( 1 + (-7.01e5 + 5.09e5i)T + (3.63e11 - 1.11e12i)T^{2} \)
59 \( 1 + (-9.53e5 + 2.93e6i)T + (-2.01e12 - 1.46e12i)T^{2} \)
61 \( 1 + (3.70e5 + 2.69e5i)T + (9.71e11 + 2.98e12i)T^{2} \)
67 \( 1 + 1.21e6T + 6.06e12T^{2} \)
71 \( 1 + (-7.21e5 - 5.23e5i)T + (2.81e12 + 8.64e12i)T^{2} \)
73 \( 1 + (9.10e5 - 2.80e6i)T + (-8.93e12 - 6.49e12i)T^{2} \)
79 \( 1 + (-5.31e6 + 3.86e6i)T + (5.93e12 - 1.82e13i)T^{2} \)
83 \( 1 + (6.31e6 + 4.58e6i)T + (8.38e12 + 2.58e13i)T^{2} \)
89 \( 1 + 1.00e7T + 4.42e13T^{2} \)
97 \( 1 + (-8.21e5 + 5.96e5i)T + (2.49e13 - 7.68e13i)T^{2} \)
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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

−15.63457373096188137229436991590, −14.33118116796205535565077338654, −12.93514849635925426676886442105, −10.77268841606561701242656646402, −10.05064837333464760348672853818, −8.413865472163620716925225932926, −7.70022766611264684582362291731, −5.87542740370419524026240057388, −3.71821723372173489358131835132, −0.68696864768446472587478397356, 1.32944091868760351682242017360, 2.71270391074470554063568901287, 5.57932713252695680380489577860, 7.70270871148807222205928257807, 8.785065506021758910416847062012, 9.940329817754633790652576716597, 11.43054928648450130475466428218, 12.22709739912058183688884995122, 13.81571658873960400047141307424, 15.18490508652341079423519585774

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