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Properties

Label 2-33-11.6-c2-0-3
Degree $2$
Conductor $33$
Sign $0.896 + 0.443i$
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  + (2.35 − 0.764i)2-s + (−1.40 − 1.01i)3-s + (1.72 − 1.25i)4-s + (−0.789 + 2.42i)5-s + (−4.07 − 1.32i)6-s + (−0.100 − 0.137i)7-s + (−2.72 + 3.74i)8-s + (0.927 + 2.85i)9-s + 6.32i·10-s + (−7.69 − 7.85i)11-s − 3.68·12-s + (18.3 − 5.95i)13-s + (−0.341 − 0.247i)14-s + (3.57 − 2.59i)15-s + (−6.17 + 19.0i)16-s + (−19.3 − 6.27i)17-s + ⋯
L(s)  = 1  + (1.17 − 0.382i)2-s + (−0.467 − 0.339i)3-s + (0.430 − 0.312i)4-s + (−0.157 + 0.485i)5-s + (−0.679 − 0.220i)6-s + (−0.0143 − 0.0196i)7-s + (−0.340 + 0.468i)8-s + (0.103 + 0.317i)9-s + 0.632i·10-s + (−0.699 − 0.714i)11-s − 0.307·12-s + (1.41 − 0.458i)13-s + (−0.0243 − 0.0177i)14-s + (0.238 − 0.173i)15-s + (−0.385 + 1.18i)16-s + (−1.13 − 0.368i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.896 + 0.443i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.896 + 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $0.896 + 0.443i$
Analytic conductor: \(0.899184\)
Root analytic conductor: \(0.948253\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :1),\ 0.896 + 0.443i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.36681 - 0.319885i\)
\(L(\frac12)\) \(\approx\) \(1.36681 - 0.319885i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.40 + 1.01i)T \)
11 \( 1 + (7.69 + 7.85i)T \)
good2 \( 1 + (-2.35 + 0.764i)T + (3.23 - 2.35i)T^{2} \)
5 \( 1 + (0.789 - 2.42i)T + (-20.2 - 14.6i)T^{2} \)
7 \( 1 + (0.100 + 0.137i)T + (-15.1 + 46.6i)T^{2} \)
13 \( 1 + (-18.3 + 5.95i)T + (136. - 99.3i)T^{2} \)
17 \( 1 + (19.3 + 6.27i)T + (233. + 169. i)T^{2} \)
19 \( 1 + (-8.57 + 11.8i)T + (-111. - 343. i)T^{2} \)
23 \( 1 - 7.74T + 529T^{2} \)
29 \( 1 + (-22.4 - 30.9i)T + (-259. + 799. i)T^{2} \)
31 \( 1 + (13.0 + 40.2i)T + (-777. + 564. i)T^{2} \)
37 \( 1 + (41.7 - 30.3i)T + (423. - 1.30e3i)T^{2} \)
41 \( 1 + (27.1 - 37.4i)T + (-519. - 1.59e3i)T^{2} \)
43 \( 1 - 59.7iT - 1.84e3T^{2} \)
47 \( 1 + (27.6 + 20.0i)T + (682. + 2.10e3i)T^{2} \)
53 \( 1 + (4.70 + 14.4i)T + (-2.27e3 + 1.65e3i)T^{2} \)
59 \( 1 + (-21.3 + 15.4i)T + (1.07e3 - 3.31e3i)T^{2} \)
61 \( 1 + (-60.2 - 19.5i)T + (3.01e3 + 2.18e3i)T^{2} \)
67 \( 1 + 2.91T + 4.48e3T^{2} \)
71 \( 1 + (-29.9 + 92.0i)T + (-4.07e3 - 2.96e3i)T^{2} \)
73 \( 1 + (10.1 + 14.0i)T + (-1.64e3 + 5.06e3i)T^{2} \)
79 \( 1 + (50.1 - 16.3i)T + (5.04e3 - 3.66e3i)T^{2} \)
83 \( 1 + (-22.3 - 7.25i)T + (5.57e3 + 4.04e3i)T^{2} \)
89 \( 1 - 97.1T + 7.92e3T^{2} \)
97 \( 1 + (15.6 + 48.1i)T + (-7.61e3 + 5.53e3i)T^{2} \)
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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

−16.13922016415717773128141460423, −15.04422405786732899281577508653, −13.53686669228741378747727530807, −13.10800397928980713579001772101, −11.49034988539668826682906232951, −10.86877691553171002372304418116, −8.499761539939320743529410002410, −6.54746715551602522310265256778, −5.11956771808791459996793652974, −3.17871944348555400207830985126, 4.03731350739414275373144785718, 5.29323744001235587798667716057, 6.69954801816886492603407261794, 8.824750448146746232837587067801, 10.52696147664727676192524409334, 12.06519145380806212146468824659, 13.04208612747892918149305690746, 14.13821606790320908306014514887, 15.63084959372411453724996420114, 15.96622690739083376284521479537

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