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Properties

Label 2-33-3.2-c8-0-4
Degree $2$
Conductor $33$
Sign $-0.977 + 0.210i$
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  + 23.5i·2-s + (−17.0 − 79.1i)3-s − 299.·4-s + 958. i·5-s + (1.86e3 − 401. i)6-s + 3.89e3·7-s − 1.03e3i·8-s + (−5.98e3 + 2.69e3i)9-s − 2.26e4·10-s − 4.41e3i·11-s + (5.10e3 + 2.37e4i)12-s − 4.31e4·13-s + 9.17e4i·14-s + (7.59e4 − 1.63e4i)15-s − 5.23e4·16-s + 3.45e4i·17-s + ⋯
L(s)  = 1  + 1.47i·2-s + (−0.210 − 0.977i)3-s − 1.17·4-s + 1.53i·5-s + (1.44 − 0.309i)6-s + 1.62·7-s − 0.253i·8-s + (−0.911 + 0.411i)9-s − 2.26·10-s − 0.301i·11-s + (0.246 + 1.14i)12-s − 1.51·13-s + 2.38i·14-s + (1.49 − 0.322i)15-s − 0.798·16-s + 0.413i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 + 0.210i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.977 + 0.210i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $-0.977 + 0.210i$
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: \((2,\ 33,\ (\ :4),\ -0.977 + 0.210i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.131884 - 1.24058i\)
\(L(\frac12)\) \(\approx\) \(0.131884 - 1.24058i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (17.0 + 79.1i)T \)
11 \( 1 + 4.41e3iT \)
good2 \( 1 - 23.5iT - 256T^{2} \)
5 \( 1 - 958. iT - 3.90e5T^{2} \)
7 \( 1 - 3.89e3T + 5.76e6T^{2} \)
13 \( 1 + 4.31e4T + 8.15e8T^{2} \)
17 \( 1 - 3.45e4iT - 6.97e9T^{2} \)
19 \( 1 + 1.30e5T + 1.69e10T^{2} \)
23 \( 1 - 2.06e5iT - 7.83e10T^{2} \)
29 \( 1 - 4.96e5iT - 5.00e11T^{2} \)
31 \( 1 - 1.29e6T + 8.52e11T^{2} \)
37 \( 1 + 2.31e6T + 3.51e12T^{2} \)
41 \( 1 - 1.91e6iT - 7.98e12T^{2} \)
43 \( 1 - 5.29e5T + 1.16e13T^{2} \)
47 \( 1 + 4.70e6iT - 2.38e13T^{2} \)
53 \( 1 + 1.81e6iT - 6.22e13T^{2} \)
59 \( 1 - 9.40e6iT - 1.46e14T^{2} \)
61 \( 1 + 7.24e6T + 1.91e14T^{2} \)
67 \( 1 - 1.43e7T + 4.06e14T^{2} \)
71 \( 1 - 1.67e7iT - 6.45e14T^{2} \)
73 \( 1 - 2.32e7T + 8.06e14T^{2} \)
79 \( 1 - 5.94e7T + 1.51e15T^{2} \)
83 \( 1 - 1.72e7iT - 2.25e15T^{2} \)
89 \( 1 - 3.40e7iT - 3.93e15T^{2} \)
97 \( 1 - 1.17e8T + 7.83e15T^{2} \)
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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

−15.12197966113377550949059980758, −14.57459224908687094965911063200, −13.82049470410866282417314614450, −11.83120610588447545821095654383, −10.74660810972441174772565235967, −8.296289184910174795181148717301, −7.43417156429977417124492200197, −6.54873267246613103880614945062, −5.13572758590351894386506758852, −2.22624037412637619459554280271, 0.52124123580684148844519540898, 2.13216973167682959189216969681, 4.48638098238551692416796245480, 4.87364275864452620533110475896, 8.345048284096258213550855401076, 9.443479069399079232184633399219, 10.59148092492279730477383221777, 11.81514582102128763476467339148, 12.42292039071173740268569424399, 14.11435065539420520415907344773

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