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Properties

Label 2-333-37.23-c2-0-9
Degree $2$
Conductor $333$
Sign $-0.974 - 0.224i$
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  + (0.829 + 3.09i)2-s + (−5.43 + 3.13i)4-s + (1.13 − 4.22i)5-s + (4.74 + 8.21i)7-s + (−5.14 − 5.14i)8-s + 14.0·10-s + 2.60i·11-s + (−4.01 + 15.0i)13-s + (−21.5 + 21.5i)14-s + (−0.877 + 1.52i)16-s + (2.20 − 0.591i)17-s + (−0.295 + 1.10i)19-s + (7.09 + 26.4i)20-s + (−8.06 + 2.16i)22-s + (−11.3 − 11.3i)23-s + ⋯
L(s)  = 1  + (0.414 + 1.54i)2-s + (−1.35 + 0.783i)4-s + (0.226 − 0.844i)5-s + (0.677 + 1.17i)7-s + (−0.643 − 0.643i)8-s + 1.40·10-s + 0.236i·11-s + (−0.309 + 1.15i)13-s + (−1.53 + 1.53i)14-s + (−0.0548 + 0.0950i)16-s + (0.129 − 0.0348i)17-s + (−0.0155 + 0.0579i)19-s + (0.354 + 1.32i)20-s + (−0.366 + 0.0982i)22-s + (−0.493 − 0.493i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 333 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 - 0.224i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 333 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.974 - 0.224i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(333\)    =    \(3^{2} \cdot 37\)
Sign: $-0.974 - 0.224i$
Analytic conductor: \(9.07359\)
Root analytic conductor: \(3.01224\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{333} (208, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 333,\ (\ :1),\ -0.974 - 0.224i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.223013 + 1.96551i\)
\(L(\frac12)\) \(\approx\) \(0.223013 + 1.96551i\)
\(L(2)\) 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 \( 1 + (-35.0 + 11.9i)T \)
good2 \( 1 + (-0.829 - 3.09i)T + (-3.46 + 2i)T^{2} \)
5 \( 1 + (-1.13 + 4.22i)T + (-21.6 - 12.5i)T^{2} \)
7 \( 1 + (-4.74 - 8.21i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 - 2.60iT - 121T^{2} \)
13 \( 1 + (4.01 - 15.0i)T + (-146. - 84.5i)T^{2} \)
17 \( 1 + (-2.20 + 0.591i)T + (250. - 144.5i)T^{2} \)
19 \( 1 + (0.295 - 1.10i)T + (-312. - 180.5i)T^{2} \)
23 \( 1 + (11.3 + 11.3i)T + 529iT^{2} \)
29 \( 1 + (7.40 - 7.40i)T - 841iT^{2} \)
31 \( 1 + (35.3 - 35.3i)T - 961iT^{2} \)
41 \( 1 + (-64.7 + 37.3i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (26.9 + 26.9i)T + 1.84e3iT^{2} \)
47 \( 1 - 35.8T + 2.20e3T^{2} \)
53 \( 1 + (52.1 - 90.2i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-48.3 + 12.9i)T + (3.01e3 - 1.74e3i)T^{2} \)
61 \( 1 + (-84.7 - 22.7i)T + (3.22e3 + 1.86e3i)T^{2} \)
67 \( 1 + (-65.9 + 38.0i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (40.9 + 70.8i)T + (-2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + 142. iT - 5.32e3T^{2} \)
79 \( 1 + (12.6 - 47.0i)T + (-5.40e3 - 3.12e3i)T^{2} \)
83 \( 1 + (30.2 - 52.3i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (-17.3 - 64.8i)T + (-6.85e3 + 3.96e3i)T^{2} \)
97 \( 1 + (-62.8 - 62.8i)T + 9.40e3iT^{2} \)
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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

−12.18417352173237411115362057775, −10.96794068736850486717584297022, −9.219867775045291608673612617702, −8.895916076810309779064994162226, −7.901296524538460089817240046004, −6.90956240881044253001178539037, −5.77948793330265909733821348331, −5.12372751874797143765584658703, −4.24765754349614362566738172959, −1.99463658698623365129335751251, 0.846631118621652184134181279315, 2.34865905870211048273129906672, 3.45524069815561336948144528907, 4.40546714118607650460226993916, 5.67124485032944919544517352121, 7.17537640174494802781481256407, 8.069702051348856607432190098520, 9.753478519957660091096498637500, 10.18662132253698786157916620236, 11.22683492007781470346522140884

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