Properties

Label 2-33-33.2-c5-0-10
Degree $2$
Conductor $33$
Sign $0.760 - 0.648i$
Analytic cond. $5.29266$
Root an. cond. $2.30057$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.95 + 9.09i)2-s + (10.1 − 11.8i)3-s + (−48.0 − 34.8i)4-s + (91.6 − 29.7i)5-s + (77.9 + 126. i)6-s + (70.5 − 97.1i)7-s + (211. − 153. i)8-s + (−38.3 − 239. i)9-s + 921. i·10-s + (−277. + 289. i)11-s + (−899. + 216. i)12-s + (390. + 126. i)13-s + (674. + 928. i)14-s + (573. − 1.38e3i)15-s + (185. + 571. i)16-s + (−39.0 − 120. i)17-s + ⋯
L(s)  = 1  + (−0.522 + 1.60i)2-s + (0.648 − 0.760i)3-s + (−1.50 − 1.09i)4-s + (1.63 − 0.532i)5-s + (0.883 + 1.44i)6-s + (0.544 − 0.749i)7-s + (1.16 − 0.849i)8-s + (−0.157 − 0.987i)9-s + 2.91i·10-s + (−0.692 + 0.721i)11-s + (−1.80 + 0.434i)12-s + (0.640 + 0.208i)13-s + (0.919 + 1.26i)14-s + (0.658 − 1.59i)15-s + (0.181 + 0.557i)16-s + (−0.0327 − 0.100i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.760 - 0.648i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.760 - 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $0.760 - 0.648i$
Analytic conductor: \(5.29266\)
Root analytic conductor: \(2.30057\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :5/2),\ 0.760 - 0.648i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.61451 + 0.594987i\)
\(L(\frac12)\) \(\approx\) \(1.61451 + 0.594987i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-10.1 + 11.8i)T \)
11 \( 1 + (277. - 289. i)T \)
good2 \( 1 + (2.95 - 9.09i)T + (-25.8 - 18.8i)T^{2} \)
5 \( 1 + (-91.6 + 29.7i)T + (2.52e3 - 1.83e3i)T^{2} \)
7 \( 1 + (-70.5 + 97.1i)T + (-5.19e3 - 1.59e4i)T^{2} \)
13 \( 1 + (-390. - 126. i)T + (3.00e5 + 2.18e5i)T^{2} \)
17 \( 1 + (39.0 + 120. i)T + (-1.14e6 + 8.34e5i)T^{2} \)
19 \( 1 + (-1.06e3 - 1.46e3i)T + (-7.65e5 + 2.35e6i)T^{2} \)
23 \( 1 + 379. iT - 6.43e6T^{2} \)
29 \( 1 + (582. + 423. i)T + (6.33e6 + 1.95e7i)T^{2} \)
31 \( 1 + (2.34e3 - 7.22e3i)T + (-2.31e7 - 1.68e7i)T^{2} \)
37 \( 1 + (9.26e3 + 6.73e3i)T + (2.14e7 + 6.59e7i)T^{2} \)
41 \( 1 + (6.79e3 - 4.93e3i)T + (3.58e7 - 1.10e8i)T^{2} \)
43 \( 1 - 7.79e3iT - 1.47e8T^{2} \)
47 \( 1 + (-1.57e3 - 2.17e3i)T + (-7.08e7 + 2.18e8i)T^{2} \)
53 \( 1 + (2.56e4 + 8.31e3i)T + (3.38e8 + 2.45e8i)T^{2} \)
59 \( 1 + (-7.74e3 + 1.06e4i)T + (-2.20e8 - 6.79e8i)T^{2} \)
61 \( 1 + (5.20e3 - 1.69e3i)T + (6.83e8 - 4.96e8i)T^{2} \)
67 \( 1 - 4.77e4T + 1.35e9T^{2} \)
71 \( 1 + (-8.27e3 + 2.68e3i)T + (1.45e9 - 1.06e9i)T^{2} \)
73 \( 1 + (8.64e3 - 1.18e4i)T + (-6.40e8 - 1.97e9i)T^{2} \)
79 \( 1 + (5.41e4 + 1.75e4i)T + (2.48e9 + 1.80e9i)T^{2} \)
83 \( 1 + (-1.54e4 - 4.75e4i)T + (-3.18e9 + 2.31e9i)T^{2} \)
89 \( 1 + 6.99e4iT - 5.58e9T^{2} \)
97 \( 1 + (5.48e3 - 1.68e4i)T + (-6.94e9 - 5.04e9i)T^{2} \)
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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

−15.98236034025590741134931065010, −14.37147993318166517134710775766, −13.93956417127282025751209998394, −12.84108719933912660500926378609, −10.06168084328770647406481238575, −8.971860455645361261354503424719, −7.81052522231118653243384934085, −6.59269317249216279499693810743, −5.27123721749936312930957698969, −1.47423523212476315297335148359, 2.00480776358747034601411836434, 3.08134745786176078446953739087, 5.42466267933240293597294070330, 8.540433964485045148530541998716, 9.459086650309257608557692175163, 10.44621439795323302565909970575, 11.29324495680219363767807995434, 13.22541403566477418541477132686, 13.89355381675160209380143825091, 15.41194362981357422407811918921

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