Properties

Label 2-33-33.32-c5-0-11
Degree $2$
Conductor $33$
Sign $0.952 - 0.304i$
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  + 8.47·2-s + (11.5 + 10.4i)3-s + 39.8·4-s − 35.7i·5-s + (98.3 + 88.3i)6-s − 11.5i·7-s + 66.9·8-s + (25.9 + 241. i)9-s − 302. i·10-s + (−202. + 346. i)11-s + (462. + 415. i)12-s − 933. i·13-s − 98.0i·14-s + (371. − 414. i)15-s − 708.·16-s − 25.0·17-s + ⋯
L(s)  = 1  + 1.49·2-s + (0.743 + 0.668i)3-s + 1.24·4-s − 0.638i·5-s + (1.11 + 1.00i)6-s − 0.0892i·7-s + 0.369·8-s + (0.106 + 0.994i)9-s − 0.957i·10-s + (−0.505 + 0.862i)11-s + (0.927 + 0.833i)12-s − 1.53i·13-s − 0.133i·14-s + (0.426 − 0.475i)15-s − 0.692·16-s − 0.0210·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.304i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.952 - 0.304i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $0.952 - 0.304i$
Analytic conductor: \(5.29266\)
Root analytic conductor: \(2.30057\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :5/2),\ 0.952 - 0.304i)\)

Particular Values

\(L(3)\) \(\approx\) \(3.66851 + 0.571388i\)
\(L(\frac12)\) \(\approx\) \(3.66851 + 0.571388i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-11.5 - 10.4i)T \)
11 \( 1 + (202. - 346. i)T \)
good2 \( 1 - 8.47T + 32T^{2} \)
5 \( 1 + 35.7iT - 3.12e3T^{2} \)
7 \( 1 + 11.5iT - 1.68e4T^{2} \)
13 \( 1 + 933. iT - 3.71e5T^{2} \)
17 \( 1 + 25.0T + 1.41e6T^{2} \)
19 \( 1 + 743. iT - 2.47e6T^{2} \)
23 \( 1 - 594. iT - 6.43e6T^{2} \)
29 \( 1 + 3.97e3T + 2.05e7T^{2} \)
31 \( 1 + 3.07e3T + 2.86e7T^{2} \)
37 \( 1 - 1.03e4T + 6.93e7T^{2} \)
41 \( 1 - 1.77e4T + 1.15e8T^{2} \)
43 \( 1 - 2.29e4iT - 1.47e8T^{2} \)
47 \( 1 - 3.40e3iT - 2.29e8T^{2} \)
53 \( 1 + 1.23e4iT - 4.18e8T^{2} \)
59 \( 1 - 3.03e4iT - 7.14e8T^{2} \)
61 \( 1 + 3.64e4iT - 8.44e8T^{2} \)
67 \( 1 + 2.85e4T + 1.35e9T^{2} \)
71 \( 1 + 5.93e4iT - 1.80e9T^{2} \)
73 \( 1 - 7.24e4iT - 2.07e9T^{2} \)
79 \( 1 + 2.69e4iT - 3.07e9T^{2} \)
83 \( 1 - 3.24e4T + 3.93e9T^{2} \)
89 \( 1 + 1.08e4iT - 5.58e9T^{2} \)
97 \( 1 + 1.05e5T + 8.58e9T^{2} \)
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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

−15.30098077045425292172443314767, −14.64697943155063364887406951477, −13.19827844722227402159526455345, −12.72285364355068766988975842112, −10.91201365185255748112576176672, −9.386041376315941467985273451917, −7.72302421343714921652172577235, −5.45100230802851041695428517323, −4.38451809607134296854856953657, −2.80871458074465628853742241098, 2.46700860963799403565050440010, 3.88573204932314259153819131485, 5.98248101187934986595930523458, 7.19643992818012241844277503536, 9.003524721172329102120785171216, 11.14799908513775198771639206131, 12.32900847048789034994823791986, 13.47546510467402095957766588375, 14.22632439654475423919925983521, 14.99930908198336919723968286981

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