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Properties

Label 2-33-11.8-c2-0-1
Degree $2$
Conductor $33$
Sign $0.537 - 0.843i$
Analytic cond. $0.899184$
Root an. cond. $0.948253$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.30 + 1.79i)2-s + (0.535 + 1.64i)3-s + (−0.280 + 0.862i)4-s + (−7.03 − 5.11i)5-s + (−2.25 + 3.10i)6-s + (6.34 + 2.06i)7-s + (6.51 − 2.11i)8-s + (−2.42 + 1.76i)9-s − 19.2i·10-s + (−8.95 + 6.38i)11-s − 1.57·12-s + (−6.55 − 9.02i)13-s + (4.56 + 14.0i)14-s + (4.65 − 14.3i)15-s + (15.2 + 11.0i)16-s + (−4.60 + 6.33i)17-s + ⋯
L(s)  = 1  + (0.651 + 0.896i)2-s + (0.178 + 0.549i)3-s + (−0.0700 + 0.215i)4-s + (−1.40 − 1.02i)5-s + (−0.375 + 0.517i)6-s + (0.906 + 0.294i)7-s + (0.814 − 0.264i)8-s + (−0.269 + 0.195i)9-s − 1.92i·10-s + (−0.813 + 0.580i)11-s − 0.130·12-s + (−0.504 − 0.694i)13-s + (0.326 + 1.00i)14-s + (0.310 − 0.954i)15-s + (0.950 + 0.690i)16-s + (−0.270 + 0.372i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.537 - 0.843i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.537 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $0.537 - 0.843i$
Analytic conductor: \(0.899184\)
Root analytic conductor: \(0.948253\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :1),\ 0.537 - 0.843i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.12231 + 0.615700i\)
\(L(\frac12)\) \(\approx\) \(1.12231 + 0.615700i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.535 - 1.64i)T \)
11 \( 1 + (8.95 - 6.38i)T \)
good2 \( 1 + (-1.30 - 1.79i)T + (-1.23 + 3.80i)T^{2} \)
5 \( 1 + (7.03 + 5.11i)T + (7.72 + 23.7i)T^{2} \)
7 \( 1 + (-6.34 - 2.06i)T + (39.6 + 28.8i)T^{2} \)
13 \( 1 + (6.55 + 9.02i)T + (-52.2 + 160. i)T^{2} \)
17 \( 1 + (4.60 - 6.33i)T + (-89.3 - 274. i)T^{2} \)
19 \( 1 + (8.02 - 2.60i)T + (292. - 212. i)T^{2} \)
23 \( 1 - 9.30T + 529T^{2} \)
29 \( 1 + (-6.82 - 2.21i)T + (680. + 494. i)T^{2} \)
31 \( 1 + (-22.1 + 16.1i)T + (296. - 913. i)T^{2} \)
37 \( 1 + (16.3 - 50.3i)T + (-1.10e3 - 804. i)T^{2} \)
41 \( 1 + (-45.5 + 14.7i)T + (1.35e3 - 988. i)T^{2} \)
43 \( 1 + 45.8iT - 1.84e3T^{2} \)
47 \( 1 + (4.96 + 15.2i)T + (-1.78e3 + 1.29e3i)T^{2} \)
53 \( 1 + (-44.4 + 32.2i)T + (868. - 2.67e3i)T^{2} \)
59 \( 1 + (22.0 - 67.7i)T + (-2.81e3 - 2.04e3i)T^{2} \)
61 \( 1 + (0.764 - 1.05i)T + (-1.14e3 - 3.53e3i)T^{2} \)
67 \( 1 + 28.9T + 4.48e3T^{2} \)
71 \( 1 + (17.8 + 12.9i)T + (1.55e3 + 4.79e3i)T^{2} \)
73 \( 1 + (119. + 38.6i)T + (4.31e3 + 3.13e3i)T^{2} \)
79 \( 1 + (-25.2 - 34.6i)T + (-1.92e3 + 5.93e3i)T^{2} \)
83 \( 1 + (-78.3 + 107. i)T + (-2.12e3 - 6.55e3i)T^{2} \)
89 \( 1 + 9.48T + 7.92e3T^{2} \)
97 \( 1 + (126. - 91.7i)T + (2.90e3 - 8.94e3i)T^{2} \)
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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

−16.24167779577459672657681094100, −15.25727223978898573880677724314, −14.97585900532639483727166245268, −13.21706670888341301065646938183, −11.99993443605833920748194167686, −10.50372829007302881375203580794, −8.450926646300029460241982554761, −7.58452472828800989253786288667, −5.20797058097258475114000588228, −4.38862001674889299598542538852, 2.80288377966965723474973235437, 4.41321585574761399572764056387, 7.20934699204005754891712598785, 8.105180702747960290062599396130, 10.83646939382016779866735816169, 11.37322380380109258347343758504, 12.43464325242000519947705812250, 13.86045515051062298504328542694, 14.72108659230017876223366964777, 16.14188457222500803951163800746

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