Properties

Label 2-33-11.3-c5-0-4
Degree $2$
Conductor $33$
Sign $0.567 - 0.823i$
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  + (3.08 − 2.23i)2-s + (2.78 + 8.55i)3-s + (−5.40 + 16.6i)4-s + (20.5 + 14.9i)5-s + (27.7 + 20.1i)6-s + (−26.9 + 82.8i)7-s + (58.2 + 179. i)8-s + (−65.5 + 47.6i)9-s + 96.8·10-s + (375. − 142. i)11-s − 157.·12-s + (270. − 196. i)13-s + (102. + 315. i)14-s + (−70.7 + 217. i)15-s + (127. + 92.6i)16-s + (−125. − 91.0i)17-s + ⋯
L(s)  = 1  + (0.544 − 0.395i)2-s + (0.178 + 0.549i)3-s + (−0.169 + 0.520i)4-s + (0.367 + 0.267i)5-s + (0.314 + 0.228i)6-s + (−0.207 + 0.638i)7-s + (0.321 + 0.990i)8-s + (−0.269 + 0.195i)9-s + 0.306·10-s + (0.935 − 0.354i)11-s − 0.315·12-s + (0.443 − 0.322i)13-s + (0.139 + 0.429i)14-s + (−0.0811 + 0.249i)15-s + (0.124 + 0.0904i)16-s + (−0.105 − 0.0763i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.567 - 0.823i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.567 - 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $0.567 - 0.823i$
Analytic conductor: \(5.29266\)
Root analytic conductor: \(2.30057\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :5/2),\ 0.567 - 0.823i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.86499 + 0.978962i\)
\(L(\frac12)\) \(\approx\) \(1.86499 + 0.978962i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.78 - 8.55i)T \)
11 \( 1 + (-375. + 142. i)T \)
good2 \( 1 + (-3.08 + 2.23i)T + (9.88 - 30.4i)T^{2} \)
5 \( 1 + (-20.5 - 14.9i)T + (965. + 2.97e3i)T^{2} \)
7 \( 1 + (26.9 - 82.8i)T + (-1.35e4 - 9.87e3i)T^{2} \)
13 \( 1 + (-270. + 196. i)T + (1.14e5 - 3.53e5i)T^{2} \)
17 \( 1 + (125. + 91.0i)T + (4.38e5 + 1.35e6i)T^{2} \)
19 \( 1 + (153. + 471. i)T + (-2.00e6 + 1.45e6i)T^{2} \)
23 \( 1 + 1.32e3T + 6.43e6T^{2} \)
29 \( 1 + (329. - 1.01e3i)T + (-1.65e7 - 1.20e7i)T^{2} \)
31 \( 1 + (-8.14e3 + 5.92e3i)T + (8.84e6 - 2.72e7i)T^{2} \)
37 \( 1 + (-1.42e3 + 4.37e3i)T + (-5.61e7 - 4.07e7i)T^{2} \)
41 \( 1 + (-4.82e3 - 1.48e4i)T + (-9.37e7 + 6.80e7i)T^{2} \)
43 \( 1 - 1.13e4T + 1.47e8T^{2} \)
47 \( 1 + (-2.16e3 - 6.65e3i)T + (-1.85e8 + 1.34e8i)T^{2} \)
53 \( 1 + (-1.16e4 + 8.47e3i)T + (1.29e8 - 3.97e8i)T^{2} \)
59 \( 1 + (-8.69e3 + 2.67e4i)T + (-5.78e8 - 4.20e8i)T^{2} \)
61 \( 1 + (4.06e4 + 2.95e4i)T + (2.60e8 + 8.03e8i)T^{2} \)
67 \( 1 + 4.85e4T + 1.35e9T^{2} \)
71 \( 1 + (-2.37e4 - 1.72e4i)T + (5.57e8 + 1.71e9i)T^{2} \)
73 \( 1 + (3.29e3 - 1.01e4i)T + (-1.67e9 - 1.21e9i)T^{2} \)
79 \( 1 + (3.56e4 - 2.59e4i)T + (9.50e8 - 2.92e9i)T^{2} \)
83 \( 1 + (-381. - 276. i)T + (1.21e9 + 3.74e9i)T^{2} \)
89 \( 1 + 4.21e4T + 5.58e9T^{2} \)
97 \( 1 + (8.66e4 - 6.29e4i)T + (2.65e9 - 8.16e9i)T^{2} \)
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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

−15.78737886851104140130083932066, −14.44921461556550175246014885808, −13.51147483046212724162412974902, −12.18818320271916856340632189202, −11.09669024458385533908539148350, −9.482460508220403226118683430719, −8.246311236977889135355377333512, −6.03365067933227296874214641303, −4.22181638765939275004817495340, −2.71056381554252999579516360065, 1.27477431240892111486023850649, 4.15310706348032457047188789733, 5.96624315055950736102852174263, 7.10052210435637617750475754870, 9.038504310435301975834927979136, 10.33641114226292226144125938259, 12.13131345087804191094285345751, 13.49316647453779966665757766910, 14.03923101813830447317277149835, 15.27873138256866092484600022298

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