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Properties

Label 2-33-3.2-c4-0-5
Degree $2$
Conductor $33$
Sign $0.981 - 0.191i$
Analytic cond. $3.41120$
Root an. cond. $1.84694$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.64i·2-s + (1.72 + 8.83i)3-s + 2.68·4-s + 34.1i·5-s + (32.2 − 6.30i)6-s + 45.8·7-s − 68.1i·8-s + (−75.0 + 30.5i)9-s + 124.·10-s − 36.4i·11-s + (4.64 + 23.7i)12-s + 161.·13-s − 167. i·14-s + (−301. + 59.0i)15-s − 205.·16-s + 35.8i·17-s + ⋯
L(s)  = 1  − 0.912i·2-s + (0.191 + 0.981i)3-s + 0.167·4-s + 1.36i·5-s + (0.895 − 0.175i)6-s + 0.934·7-s − 1.06i·8-s + (−0.926 + 0.376i)9-s + 1.24·10-s − 0.301i·11-s + (0.0322 + 0.164i)12-s + 0.952·13-s − 0.852i·14-s + (−1.34 + 0.262i)15-s − 0.803·16-s + 0.124i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.191i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.981 - 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $0.981 - 0.191i$
Analytic conductor: \(3.41120\)
Root analytic conductor: \(1.84694\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :2),\ 0.981 - 0.191i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.69984 + 0.164632i\)
\(L(\frac12)\) \(\approx\) \(1.69984 + 0.164632i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.72 - 8.83i)T \)
11 \( 1 + 36.4iT \)
good2 \( 1 + 3.64iT - 16T^{2} \)
5 \( 1 - 34.1iT - 625T^{2} \)
7 \( 1 - 45.8T + 2.40e3T^{2} \)
13 \( 1 - 161.T + 2.85e4T^{2} \)
17 \( 1 - 35.8iT - 8.35e4T^{2} \)
19 \( 1 + 549.T + 1.30e5T^{2} \)
23 \( 1 + 363. iT - 2.79e5T^{2} \)
29 \( 1 + 969. iT - 7.07e5T^{2} \)
31 \( 1 - 1.81e3T + 9.23e5T^{2} \)
37 \( 1 + 1.59e3T + 1.87e6T^{2} \)
41 \( 1 + 917. iT - 2.82e6T^{2} \)
43 \( 1 + 542.T + 3.41e6T^{2} \)
47 \( 1 + 1.64e3iT - 4.87e6T^{2} \)
53 \( 1 - 3.06e3iT - 7.89e6T^{2} \)
59 \( 1 - 4.25e3iT - 1.21e7T^{2} \)
61 \( 1 - 2.62e3T + 1.38e7T^{2} \)
67 \( 1 - 3.87e3T + 2.01e7T^{2} \)
71 \( 1 - 2.89e3iT - 2.54e7T^{2} \)
73 \( 1 + 4.85e3T + 2.83e7T^{2} \)
79 \( 1 - 7.17e3T + 3.89e7T^{2} \)
83 \( 1 - 8.35e3iT - 4.74e7T^{2} \)
89 \( 1 + 8.54e3iT - 6.27e7T^{2} \)
97 \( 1 + 1.08e4T + 8.85e7T^{2} \)
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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

−15.67066297753761982678682690492, −14.89382986596939026790408599174, −13.73632397268419881857056930356, −11.71475974714574010973290517059, −10.81442999955468437146287575402, −10.29915008318851130575457176464, −8.408012983159543007362565168062, −6.40454546369455813091916543456, −4.01121938847311481563989521074, −2.52729682595026094871170167433, 1.59853907764022310733962055026, 5.05722884183438551097383424165, 6.52528552088797013105789927597, 8.097124965471871411988057231225, 8.644429458060921778483534589698, 11.25431202834835792618968726775, 12.41843410149773512966153919144, 13.60031978430914522665229451375, 14.77277016793859423438334625846, 15.99664947522209879193986399462

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