Properties

Degree $2$
Conductor $33$
Sign $0.514 + 0.857i$
Motivic weight $4$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.00i·2-s + 5.19·3-s + 6.98·4-s + 8.72·5-s − 15.6i·6-s − 1.45i·7-s − 69.0i·8-s + 27·9-s − 26.1i·10-s + (−62.2 − 103. i)11-s + 36.2·12-s + 162. i·13-s − 4.37·14-s + 45.3·15-s − 95.4·16-s + 189. i·17-s + ⋯
L(s)  = 1  − 0.750i·2-s + 0.577·3-s + 0.436·4-s + 0.349·5-s − 0.433i·6-s − 0.0297i·7-s − 1.07i·8-s + 0.333·9-s − 0.261i·10-s + (−0.514 − 0.857i)11-s + 0.252·12-s + 0.959i·13-s − 0.0223·14-s + 0.201·15-s − 0.372·16-s + 0.656i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.514 + 0.857i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.514 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $0.514 + 0.857i$
Motivic weight: \(4\)
Character: $\chi_{33} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 33,\ (\ :2),\ 0.514 + 0.857i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.65765 - 0.938618i\)
\(L(\frac12)\) \(\approx\) \(1.65765 - 0.938618i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 5.19T \)
11 \( 1 + (62.2 + 103. i)T \)
good2 \( 1 + 3.00iT - 16T^{2} \)
5 \( 1 - 8.72T + 625T^{2} \)
7 \( 1 + 1.45iT - 2.40e3T^{2} \)
13 \( 1 - 162. iT - 2.85e4T^{2} \)
17 \( 1 - 189. iT - 8.35e4T^{2} \)
19 \( 1 - 590. iT - 1.30e5T^{2} \)
23 \( 1 + 12.8T + 2.79e5T^{2} \)
29 \( 1 + 282. iT - 7.07e5T^{2} \)
31 \( 1 + 304.T + 9.23e5T^{2} \)
37 \( 1 - 464.T + 1.87e6T^{2} \)
41 \( 1 - 1.19e3iT - 2.82e6T^{2} \)
43 \( 1 + 1.59e3iT - 3.41e6T^{2} \)
47 \( 1 + 1.82e3T + 4.87e6T^{2} \)
53 \( 1 + 4.02e3T + 7.89e6T^{2} \)
59 \( 1 + 1.48e3T + 1.21e7T^{2} \)
61 \( 1 + 356. iT - 1.38e7T^{2} \)
67 \( 1 - 8.25e3T + 2.01e7T^{2} \)
71 \( 1 - 7.97e3T + 2.54e7T^{2} \)
73 \( 1 + 5.78e3iT - 2.83e7T^{2} \)
79 \( 1 + 1.13e4iT - 3.89e7T^{2} \)
83 \( 1 + 5.44e3iT - 4.74e7T^{2} \)
89 \( 1 - 7.33e3T + 6.27e7T^{2} \)
97 \( 1 + 1.12e4T + 8.85e7T^{2} \)
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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

−15.78949808973298148056713659899, −14.39094497608133361991448093851, −13.24825416025651609182854248724, −12.00431889085725622890929342719, −10.72051355415894645685073972093, −9.599931194231446892773699667472, −7.966346255477406559834280129232, −6.20253280418315533846674063426, −3.64317577682149021949781282247, −1.89425696227464918216551570556, 2.53207832946159250583868089707, 5.20318308081748477483932520352, 6.92162165236381806645438406085, 8.022268840493014073263134241651, 9.597679962531862953622097647213, 11.08529950120959052686874962649, 12.74767632722944540291109899022, 13.99095912921206215569528896842, 15.23507768548708675379368248737, 15.78317962625631955262862597002

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