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Properties

Label 2-33-33.32-c3-0-5
Degree $2$
Conductor $33$
Sign $0.942 - 0.333i$
Analytic cond. $1.94706$
Root an. cond. $1.39537$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.48·2-s + (−2.57 + 4.51i)3-s + 12.1·4-s − 12.2i·5-s + (−11.5 + 20.2i)6-s + 14.5i·7-s + 18.6·8-s + (−13.7 − 23.2i)9-s − 55.0i·10-s + (−27.6 − 23.8i)11-s + (−31.3 + 54.8i)12-s − 14.5i·13-s + 65.2i·14-s + (55.3 + 31.5i)15-s − 13.5·16-s + 92.5·17-s + ⋯
L(s)  = 1  + 1.58·2-s + (−0.495 + 0.868i)3-s + 1.51·4-s − 1.09i·5-s + (−0.786 + 1.37i)6-s + 0.784i·7-s + 0.823·8-s + (−0.508 − 0.861i)9-s − 1.74i·10-s + (−0.756 − 0.653i)11-s + (−0.753 + 1.31i)12-s − 0.310i·13-s + 1.24i·14-s + (0.952 + 0.543i)15-s − 0.211·16-s + 1.31·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.333i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.942 - 0.333i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $0.942 - 0.333i$
Analytic conductor: \(1.94706\)
Root analytic conductor: \(1.39537\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :3/2),\ 0.942 - 0.333i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.17130 + 0.372709i\)
\(L(\frac12)\) \(\approx\) \(2.17130 + 0.372709i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.57 - 4.51i)T \)
11 \( 1 + (27.6 + 23.8i)T \)
good2 \( 1 - 4.48T + 8T^{2} \)
5 \( 1 + 12.2iT - 125T^{2} \)
7 \( 1 - 14.5iT - 343T^{2} \)
13 \( 1 + 14.5iT - 2.19e3T^{2} \)
17 \( 1 - 92.5T + 4.91e3T^{2} \)
19 \( 1 - 162. iT - 6.85e3T^{2} \)
23 \( 1 - 23.8iT - 1.21e4T^{2} \)
29 \( 1 - 52.5T + 2.43e4T^{2} \)
31 \( 1 - 82.5T + 2.97e4T^{2} \)
37 \( 1 + 276.T + 5.06e4T^{2} \)
41 \( 1 + 175.T + 6.89e4T^{2} \)
43 \( 1 + 353. iT - 7.95e4T^{2} \)
47 \( 1 + 387. iT - 1.03e5T^{2} \)
53 \( 1 - 79.9iT - 1.48e5T^{2} \)
59 \( 1 - 291. iT - 2.05e5T^{2} \)
61 \( 1 + 147. iT - 2.26e5T^{2} \)
67 \( 1 - 201.T + 3.00e5T^{2} \)
71 \( 1 + 424. iT - 3.57e5T^{2} \)
73 \( 1 - 461. iT - 3.89e5T^{2} \)
79 \( 1 + 122. iT - 4.93e5T^{2} \)
83 \( 1 + 151.T + 5.71e5T^{2} \)
89 \( 1 - 764. iT - 7.04e5T^{2} \)
97 \( 1 - 218.T + 9.12e5T^{2} \)
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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

−16.02410616265353875880125678736, −15.11588145174491211076432998434, −13.90428271633135837726548231764, −12.42177565413194446952799083117, −11.98764872644779396317483242421, −10.26859422193269043763132758305, −8.530754852554832567482731034359, −5.73823296033989153283522557238, −5.22037423818785453677103945367, −3.53464179954195476101783881965, 2.83146810916314971674465148256, 4.93641389081584091403408843223, 6.56183360819712869106845990079, 7.38046048772218343155171396972, 10.52167097174116646014062681510, 11.55181965309868582957256501666, 12.75488244084321696274557941812, 13.71052329994984533347520918102, 14.53073433645075315440071954812, 15.78871972066501659689493583451

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