Properties

Label 2-33-11.8-c2-0-3
Degree $2$
Conductor $33$
Sign $0.332 + 0.943i$
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  + (−0.577 − 0.794i)2-s + (−0.535 − 1.64i)3-s + (0.937 − 2.88i)4-s + (−0.321 − 0.233i)5-s + (−1.00 + 1.37i)6-s + (6.87 + 2.23i)7-s + (−6.57 + 2.13i)8-s + (−2.42 + 1.76i)9-s + 0.390i·10-s + (−0.495 + 10.9i)11-s − 5.25·12-s + (7.14 + 9.83i)13-s + (−2.19 − 6.75i)14-s + (−0.212 + 0.654i)15-s + (−4.32 − 3.14i)16-s + (12.2 − 16.9i)17-s + ⋯
L(s)  = 1  + (−0.288 − 0.397i)2-s + (−0.178 − 0.549i)3-s + (0.234 − 0.721i)4-s + (−0.0643 − 0.0467i)5-s + (−0.166 + 0.229i)6-s + (0.981 + 0.319i)7-s + (−0.821 + 0.266i)8-s + (−0.269 + 0.195i)9-s + 0.0390i·10-s + (−0.0450 + 0.998i)11-s − 0.438·12-s + (0.549 + 0.756i)13-s + (−0.156 − 0.482i)14-s + (−0.0141 + 0.0436i)15-s + (−0.270 − 0.196i)16-s + (0.722 − 0.995i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.332 + 0.943i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.332 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $0.332 + 0.943i$
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: \((2,\ 33,\ (\ :1),\ 0.332 + 0.943i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.740072 - 0.523967i\)
\(L(\frac12)\) \(\approx\) \(0.740072 - 0.523967i\)
\(L(2)\) 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 \( 1 + (0.495 - 10.9i)T \)
good2 \( 1 + (0.577 + 0.794i)T + (-1.23 + 3.80i)T^{2} \)
5 \( 1 + (0.321 + 0.233i)T + (7.72 + 23.7i)T^{2} \)
7 \( 1 + (-6.87 - 2.23i)T + (39.6 + 28.8i)T^{2} \)
13 \( 1 + (-7.14 - 9.83i)T + (-52.2 + 160. i)T^{2} \)
17 \( 1 + (-12.2 + 16.9i)T + (-89.3 - 274. i)T^{2} \)
19 \( 1 + (10.5 - 3.43i)T + (292. - 212. i)T^{2} \)
23 \( 1 + 5.92T + 529T^{2} \)
29 \( 1 + (-23.7 - 7.71i)T + (680. + 494. i)T^{2} \)
31 \( 1 + (48.2 - 35.0i)T + (296. - 913. i)T^{2} \)
37 \( 1 + (1.84 - 5.67i)T + (-1.10e3 - 804. i)T^{2} \)
41 \( 1 + (49.4 - 16.0i)T + (1.35e3 - 988. i)T^{2} \)
43 \( 1 - 17.6iT - 1.84e3T^{2} \)
47 \( 1 + (17.1 + 52.6i)T + (-1.78e3 + 1.29e3i)T^{2} \)
53 \( 1 + (-76.5 + 55.6i)T + (868. - 2.67e3i)T^{2} \)
59 \( 1 + (7.75 - 23.8i)T + (-2.81e3 - 2.04e3i)T^{2} \)
61 \( 1 + (-19.6 + 27.0i)T + (-1.14e3 - 3.53e3i)T^{2} \)
67 \( 1 - 94.6T + 4.48e3T^{2} \)
71 \( 1 + (66.9 + 48.6i)T + (1.55e3 + 4.79e3i)T^{2} \)
73 \( 1 + (44.2 + 14.3i)T + (4.31e3 + 3.13e3i)T^{2} \)
79 \( 1 + (-35.4 - 48.7i)T + (-1.92e3 + 5.93e3i)T^{2} \)
83 \( 1 + (88.5 - 121. i)T + (-2.12e3 - 6.55e3i)T^{2} \)
89 \( 1 - 134.T + 7.92e3T^{2} \)
97 \( 1 + (30.4 - 22.1i)T + (2.90e3 - 8.94e3i)T^{2} \)
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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

−16.32224057569001869448386475124, −14.90399131987686252622655783461, −14.03765389728829146052387489484, −12.22084412148722984241309718480, −11.41286281683644293628902010613, −10.07277072306588281230130397167, −8.548629608919472989713084147098, −6.83567959305542649579598809256, −5.14317502831064095892490996680, −1.84109453060867794225412917651, 3.69790791199141113395245674101, 5.83942512868137111323951083051, 7.75997157248207678811117759086, 8.704578937130054583066745082693, 10.61500599303428760890315780757, 11.59953947619937101808307474943, 13.11312325713278746898401700556, 14.67177978831064240846399255129, 15.70946913340282845276138603169, 16.83841470540380427412511669712

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