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Properties

Label 2-33-11.8-c2-0-2
Degree $2$
Conductor $33$
Sign $0.596 - 0.802i$
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  + (1.69 + 2.33i)2-s + (−0.535 − 1.64i)3-s + (−1.33 + 4.10i)4-s + (0.356 + 0.259i)5-s + (2.93 − 4.04i)6-s + (−10.0 − 3.27i)7-s + (−0.881 + 0.286i)8-s + (−2.42 + 1.76i)9-s + 1.27i·10-s + (9.39 − 5.72i)11-s + 7.48·12-s + (3.78 + 5.21i)13-s + (−9.43 − 29.0i)14-s + (0.236 − 0.726i)15-s + (11.8 + 8.58i)16-s + (−11.1 + 15.2i)17-s + ⋯
L(s)  = 1  + (0.847 + 1.16i)2-s + (−0.178 − 0.549i)3-s + (−0.333 + 1.02i)4-s + (0.0713 + 0.0518i)5-s + (0.489 − 0.673i)6-s + (−1.43 − 0.467i)7-s + (−0.110 + 0.0358i)8-s + (−0.269 + 0.195i)9-s + 0.127i·10-s + (0.853 − 0.520i)11-s + 0.623·12-s + (0.291 + 0.401i)13-s + (−0.673 − 2.07i)14-s + (0.0157 − 0.0484i)15-s + (0.738 + 0.536i)16-s + (−0.653 + 0.899i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.596 - 0.802i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.596 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $0.596 - 0.802i$
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: \((2,\ 33,\ (\ :1),\ 0.596 - 0.802i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.18359 + 0.594573i\)
\(L(\frac12)\) \(\approx\) \(1.18359 + 0.594573i\)
\(L(2)\) 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 \( 1 + (-9.39 + 5.72i)T \)
good2 \( 1 + (-1.69 - 2.33i)T + (-1.23 + 3.80i)T^{2} \)
5 \( 1 + (-0.356 - 0.259i)T + (7.72 + 23.7i)T^{2} \)
7 \( 1 + (10.0 + 3.27i)T + (39.6 + 28.8i)T^{2} \)
13 \( 1 + (-3.78 - 5.21i)T + (-52.2 + 160. i)T^{2} \)
17 \( 1 + (11.1 - 15.2i)T + (-89.3 - 274. i)T^{2} \)
19 \( 1 + (26.1 - 8.51i)T + (292. - 212. i)T^{2} \)
23 \( 1 - 6.29T + 529T^{2} \)
29 \( 1 + (-42.0 - 13.6i)T + (680. + 494. i)T^{2} \)
31 \( 1 + (-21.9 + 15.9i)T + (296. - 913. i)T^{2} \)
37 \( 1 + (0.263 - 0.811i)T + (-1.10e3 - 804. i)T^{2} \)
41 \( 1 + (12.5 - 4.08i)T + (1.35e3 - 988. i)T^{2} \)
43 \( 1 + 68.8iT - 1.84e3T^{2} \)
47 \( 1 + (4.98 + 15.3i)T + (-1.78e3 + 1.29e3i)T^{2} \)
53 \( 1 + (31.2 - 22.7i)T + (868. - 2.67e3i)T^{2} \)
59 \( 1 + (-23.7 + 73.1i)T + (-2.81e3 - 2.04e3i)T^{2} \)
61 \( 1 + (16.8 - 23.1i)T + (-1.14e3 - 3.53e3i)T^{2} \)
67 \( 1 + 78.0T + 4.48e3T^{2} \)
71 \( 1 + (-21.9 - 15.9i)T + (1.55e3 + 4.79e3i)T^{2} \)
73 \( 1 + (12.3 + 4.01i)T + (4.31e3 + 3.13e3i)T^{2} \)
79 \( 1 + (-55.4 - 76.2i)T + (-1.92e3 + 5.93e3i)T^{2} \)
83 \( 1 + (68.1 - 93.8i)T + (-2.12e3 - 6.55e3i)T^{2} \)
89 \( 1 - 65.2T + 7.92e3T^{2} \)
97 \( 1 + (-56.8 + 41.3i)T + (2.90e3 - 8.94e3i)T^{2} \)
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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

−16.57539034900768403926458218204, −15.53115258732037782323928883163, −14.17823625767991102919499727768, −13.34458238843153737714162883648, −12.37066872794398636056507051358, −10.43857692740224406533813066854, −8.491210640077101886831145468031, −6.61489017001971884450165077492, −6.30764533389858119488671588960, −4.00793211295959677431621416426, 2.97880469580794270112126191909, 4.54428452289288398947467468480, 6.39942647127676600487466719260, 9.171342954187649675674775863319, 10.25313053803803006779128263841, 11.55610806604352928207714658121, 12.60516800727221724285998331478, 13.52296833499725197867465850032, 15.01942870617981054049362504729, 16.14022661673539454656231398788

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