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Properties

Label 2-33-3.2-c4-0-3
Degree $2$
Conductor $33$
Sign $0.478 - 0.878i$
Analytic cond. $3.41120$
Root an. cond. $1.84694$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.31i·2-s + (−7.90 − 4.30i)3-s + 14.2·4-s + 39.9i·5-s + (5.65 − 10.3i)6-s + 53.0·7-s + 39.8i·8-s + (43.9 + 68.0i)9-s − 52.5·10-s + 36.4i·11-s + (−112. − 61.3i)12-s − 159.·13-s + 69.7i·14-s + (171. − 315. i)15-s + 175.·16-s − 335. i·17-s + ⋯
L(s)  = 1  + 0.328i·2-s + (−0.878 − 0.478i)3-s + 0.891·4-s + 1.59i·5-s + (0.157 − 0.288i)6-s + 1.08·7-s + 0.622i·8-s + (0.542 + 0.839i)9-s − 0.525·10-s + 0.301i·11-s + (−0.783 − 0.426i)12-s − 0.940·13-s + 0.355i·14-s + (0.763 − 1.40i)15-s + 0.687·16-s − 1.16i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.478 - 0.878i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.478 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $0.478 - 0.878i$
Analytic conductor: \(3.41120\)
Root analytic conductor: \(1.84694\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :2),\ 0.478 - 0.878i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.18247 + 0.702693i\)
\(L(\frac12)\) \(\approx\) \(1.18247 + 0.702693i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (7.90 + 4.30i)T \)
11 \( 1 - 36.4iT \)
good2 \( 1 - 1.31iT - 16T^{2} \)
5 \( 1 - 39.9iT - 625T^{2} \)
7 \( 1 - 53.0T + 2.40e3T^{2} \)
13 \( 1 + 159.T + 2.85e4T^{2} \)
17 \( 1 + 335. iT - 8.35e4T^{2} \)
19 \( 1 - 218.T + 1.30e5T^{2} \)
23 \( 1 + 121. iT - 2.79e5T^{2} \)
29 \( 1 - 537. iT - 7.07e5T^{2} \)
31 \( 1 + 523.T + 9.23e5T^{2} \)
37 \( 1 - 363.T + 1.87e6T^{2} \)
41 \( 1 + 3.11e3iT - 2.82e6T^{2} \)
43 \( 1 - 2.84e3T + 3.41e6T^{2} \)
47 \( 1 + 2.91e3iT - 4.87e6T^{2} \)
53 \( 1 + 2.68e3iT - 7.89e6T^{2} \)
59 \( 1 - 2.72e3iT - 1.21e7T^{2} \)
61 \( 1 - 5.17e3T + 1.38e7T^{2} \)
67 \( 1 + 7.59e3T + 2.01e7T^{2} \)
71 \( 1 - 3.60e3iT - 2.54e7T^{2} \)
73 \( 1 + 2.83e3T + 2.83e7T^{2} \)
79 \( 1 - 4.95e3T + 3.89e7T^{2} \)
83 \( 1 + 5.21e3iT - 4.74e7T^{2} \)
89 \( 1 - 3.95e3iT - 6.27e7T^{2} \)
97 \( 1 + 1.41e4T + 8.85e7T^{2} \)
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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

−16.16882019456984042823548654493, −14.92646309821787382440889965566, −14.12242766954276103471148043659, −12.02670122767730857566462806947, −11.25982170267356546316748438088, −10.36394808652458326557631097130, −7.46258708004974431915053788522, −7.01468867318728237964151482685, −5.39635331582430140402910459999, −2.32120308897522498957968358141, 1.27901584887210153739637928356, 4.48766759596913104322385669418, 5.77760998896238636250723146291, 7.86018842841904250313439050668, 9.552026279816844183209475759712, 10.99026108374920452813166080149, 11.94585741982247391983938866873, 12.78029840562676933205221225891, 14.86163680490878895417312898648, 16.03101386854667359680425622441

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