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Properties

Label 2-33-3.2-c2-0-3
Degree $2$
Conductor $33$
Sign $0.998 + 0.0620i$
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  + 0.792i·2-s + (0.186 − 2.99i)3-s + 3.37·4-s + 2.52i·5-s + (2.37 + 0.147i)6-s − 4.74·7-s + 5.84i·8-s + (−8.93 − 1.11i)9-s − 2·10-s − 3.31i·11-s + (0.627 − 10.0i)12-s − 13.4·13-s − 3.75i·14-s + (7.55 + 0.469i)15-s + 8.86·16-s + 22.6i·17-s + ⋯
L(s)  = 1  + 0.396i·2-s + (0.0620 − 0.998i)3-s + 0.843·4-s + 0.504i·5-s + (0.395 + 0.0245i)6-s − 0.677·7-s + 0.730i·8-s + (−0.992 − 0.123i)9-s − 0.200·10-s − 0.301i·11-s + (0.0523 − 0.841i)12-s − 1.03·13-s − 0.268i·14-s + (0.503 + 0.0313i)15-s + 0.553·16-s + 1.33i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0620i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.998 + 0.0620i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $0.998 + 0.0620i$
Analytic conductor: \(0.899184\)
Root analytic conductor: \(0.948253\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :1),\ 0.998 + 0.0620i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.10118 - 0.0341953i\)
\(L(\frac12)\) \(\approx\) \(1.10118 - 0.0341953i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.186 + 2.99i)T \)
11 \( 1 + 3.31iT \)
good2 \( 1 - 0.792iT - 4T^{2} \)
5 \( 1 - 2.52iT - 25T^{2} \)
7 \( 1 + 4.74T + 49T^{2} \)
13 \( 1 + 13.4T + 169T^{2} \)
17 \( 1 - 22.6iT - 289T^{2} \)
19 \( 1 - 8.23T + 361T^{2} \)
23 \( 1 + 30.2iT - 529T^{2} \)
29 \( 1 + 53.6iT - 841T^{2} \)
31 \( 1 - 25.8T + 961T^{2} \)
37 \( 1 + 42.6T + 1.36e3T^{2} \)
41 \( 1 - 30.8iT - 1.68e3T^{2} \)
43 \( 1 - 35.4T + 1.84e3T^{2} \)
47 \( 1 + 31.2iT - 2.20e3T^{2} \)
53 \( 1 - 9.91iT - 2.80e3T^{2} \)
59 \( 1 - 61.5iT - 3.48e3T^{2} \)
61 \( 1 - 51.4T + 3.72e3T^{2} \)
67 \( 1 - 34.3T + 4.48e3T^{2} \)
71 \( 1 - 14.5iT - 5.04e3T^{2} \)
73 \( 1 + 88.2T + 5.32e3T^{2} \)
79 \( 1 + 93.6T + 6.24e3T^{2} \)
83 \( 1 + 34.1iT - 6.88e3T^{2} \)
89 \( 1 - 143. iT - 7.92e3T^{2} \)
97 \( 1 + 40.2T + 9.40e3T^{2} \)
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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

−16.62600890371678261941897494605, −15.20080683136179591897585856652, −14.25500112917077810433131747523, −12.78436128617951607603022551015, −11.74688001601902710871222810101, −10.34274441959044053426000489874, −8.251674173479031250181188277524, −6.98003003155221187853410198895, −6.06958190876563003013349839308, −2.65193927166806668305012795342, 3.09730072256132962541140336218, 5.15317017260276807481651970301, 7.15675360019047296817432946775, 9.244205530794933589757914979846, 10.16038445364951601916138645488, 11.51813828637949980142045169356, 12.57362824779418348116068151967, 14.29191837364684090954099983822, 15.69281483382534382404032766909, 16.20621875697739286340320444872

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