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Properties

Label 2-333-37.6-c2-0-22
Degree $2$
Conductor $333$
Sign $0.491 + 0.871i$
Analytic cond. $9.07359$
Root an. cond. $3.01224$
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  + (2.15 − 2.15i)2-s − 5.31i·4-s + (2.97 + 2.97i)5-s + 8.27·7-s + (−2.82 − 2.82i)8-s + 12.8·10-s + 6.70i·11-s + (4.81 + 4.81i)13-s + (17.8 − 17.8i)14-s + 9.04·16-s + (−8.97 − 8.97i)17-s + (−2.58 − 2.58i)19-s + (15.8 − 15.8i)20-s + (14.4 + 14.4i)22-s + (−22.5 − 22.5i)23-s + ⋯
L(s)  = 1  + (1.07 − 1.07i)2-s − 1.32i·4-s + (0.595 + 0.595i)5-s + 1.18·7-s + (−0.353 − 0.353i)8-s + 1.28·10-s + 0.609i·11-s + (0.370 + 0.370i)13-s + (1.27 − 1.27i)14-s + 0.565·16-s + (−0.527 − 0.527i)17-s + (−0.136 − 0.136i)19-s + (0.791 − 0.791i)20-s + (0.657 + 0.657i)22-s + (−0.982 − 0.982i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 333 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.491 + 0.871i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 333 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.491 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(333\)    =    \(3^{2} \cdot 37\)
Sign: $0.491 + 0.871i$
Analytic conductor: \(9.07359\)
Root analytic conductor: \(3.01224\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{333} (154, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 333,\ (\ :1),\ 0.491 + 0.871i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.06299 - 1.78937i\)
\(L(\frac12)\) \(\approx\) \(3.06299 - 1.78937i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
37 \( 1 + (-6.96 + 36.3i)T \)
good2 \( 1 + (-2.15 + 2.15i)T - 4iT^{2} \)
5 \( 1 + (-2.97 - 2.97i)T + 25iT^{2} \)
7 \( 1 - 8.27T + 49T^{2} \)
11 \( 1 - 6.70iT - 121T^{2} \)
13 \( 1 + (-4.81 - 4.81i)T + 169iT^{2} \)
17 \( 1 + (8.97 + 8.97i)T + 289iT^{2} \)
19 \( 1 + (2.58 + 2.58i)T + 361iT^{2} \)
23 \( 1 + (22.5 + 22.5i)T + 529iT^{2} \)
29 \( 1 + (22.9 - 22.9i)T - 841iT^{2} \)
31 \( 1 + (7.78 - 7.78i)T - 961iT^{2} \)
41 \( 1 - 2.70iT - 1.68e3T^{2} \)
43 \( 1 + (0.467 + 0.467i)T + 1.84e3iT^{2} \)
47 \( 1 - 4.93T + 2.20e3T^{2} \)
53 \( 1 + 21.4T + 2.80e3T^{2} \)
59 \( 1 + (15.2 + 15.2i)T + 3.48e3iT^{2} \)
61 \( 1 + (53.6 - 53.6i)T - 3.72e3iT^{2} \)
67 \( 1 - 64.0iT - 4.48e3T^{2} \)
71 \( 1 - 79.3T + 5.04e3T^{2} \)
73 \( 1 - 80.3iT - 5.32e3T^{2} \)
79 \( 1 + (-30.6 - 30.6i)T + 6.24e3iT^{2} \)
83 \( 1 + 144.T + 6.88e3T^{2} \)
89 \( 1 + (-105. + 105. i)T - 7.92e3iT^{2} \)
97 \( 1 + (57.6 + 57.6i)T + 9.40e3iT^{2} \)
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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

−11.18451206054073772019096890426, −10.71696539622588304316412299386, −9.741398502068330724681868243020, −8.466669365018763459902804331758, −7.19314307420136524215487857899, −5.95403679367401565513627654369, −4.87080540290215583020206740879, −4.05747961493266663682628078872, −2.53838213745065064448792414860, −1.72954108750755805592409235230, 1.63938007563275322328534895300, 3.69113025045557394661535882157, 4.76529029589090176938337338213, 5.60255415595004417666390093249, 6.27250252181616374116755210560, 7.71814570462794168085312662119, 8.254655736436496832150618561529, 9.453644913326270503119378764807, 10.76460472677939507003845417378, 11.70060252356072182720496271926

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