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Properties

Label 2-333-37.14-c2-0-3
Degree $2$
Conductor $333$
Sign $-0.614 - 0.789i$
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  + (−3.24 + 0.868i)2-s + (6.28 − 3.62i)4-s + (−4.12 − 1.10i)5-s + (−1.96 − 3.40i)7-s + (−7.72 + 7.72i)8-s + 14.3·10-s − 8.41i·11-s + (−9.09 − 2.43i)13-s + (9.33 + 9.33i)14-s + (3.82 − 6.62i)16-s + (2.75 + 10.2i)17-s + (0.967 + 0.259i)19-s + (−29.9 + 8.02i)20-s + (7.30 + 27.2i)22-s + (20.6 − 20.6i)23-s + ⋯
L(s)  = 1  + (−1.62 + 0.434i)2-s + (1.57 − 0.907i)4-s + (−0.825 − 0.221i)5-s + (−0.281 − 0.486i)7-s + (−0.966 + 0.966i)8-s + 1.43·10-s − 0.764i·11-s + (−0.699 − 0.187i)13-s + (0.666 + 0.666i)14-s + (0.238 − 0.413i)16-s + (0.161 + 0.604i)17-s + (0.0509 + 0.0136i)19-s + (−1.49 + 0.401i)20-s + (0.332 + 1.23i)22-s + (0.898 − 0.898i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 333 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.614 - 0.789i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 333 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.614 - 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(333\)    =    \(3^{2} \cdot 37\)
Sign: $-0.614 - 0.789i$
Analytic conductor: \(9.07359\)
Root analytic conductor: \(3.01224\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{333} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 333,\ (\ :1),\ -0.614 - 0.789i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0892138 + 0.182435i\)
\(L(\frac12)\) \(\approx\) \(0.0892138 + 0.182435i\)
\(L(2)\) 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 \( 1 + (6.94 - 36.3i)T \)
good2 \( 1 + (3.24 - 0.868i)T + (3.46 - 2i)T^{2} \)
5 \( 1 + (4.12 + 1.10i)T + (21.6 + 12.5i)T^{2} \)
7 \( 1 + (1.96 + 3.40i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + 8.41iT - 121T^{2} \)
13 \( 1 + (9.09 + 2.43i)T + (146. + 84.5i)T^{2} \)
17 \( 1 + (-2.75 - 10.2i)T + (-250. + 144.5i)T^{2} \)
19 \( 1 + (-0.967 - 0.259i)T + (312. + 180.5i)T^{2} \)
23 \( 1 + (-20.6 + 20.6i)T - 529iT^{2} \)
29 \( 1 + (-24.6 - 24.6i)T + 841iT^{2} \)
31 \( 1 + (-20.3 - 20.3i)T + 961iT^{2} \)
41 \( 1 + (64.7 - 37.3i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (14.4 - 14.4i)T - 1.84e3iT^{2} \)
47 \( 1 + 85.5T + 2.20e3T^{2} \)
53 \( 1 + (41.2 - 71.5i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-12.3 - 46.0i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (-8.49 + 31.7i)T + (-3.22e3 - 1.86e3i)T^{2} \)
67 \( 1 + (61.5 - 35.5i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (-38.0 - 65.9i)T + (-2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 - 66.9iT - 5.32e3T^{2} \)
79 \( 1 + (-76.1 - 20.4i)T + (5.40e3 + 3.12e3i)T^{2} \)
83 \( 1 + (-27.9 + 48.3i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (24.3 - 6.51i)T + (6.85e3 - 3.96e3i)T^{2} \)
97 \( 1 + (-23.4 + 23.4i)T - 9.40e3iT^{2} \)
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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

−11.34767405179360299287361441282, −10.44525688169966524187209318403, −9.832853630053599150342161452967, −8.499606533238560497476273322001, −8.263727515719252694094500661392, −7.11340804134238584615171343201, −6.39650100765861201388519984567, −4.75116064379012191562362299685, −3.14815198051086980715347808131, −1.11184757686803242147905207578, 0.18909044323931645798627363857, 2.03113005163870643309257762914, 3.27390543151202741663661978806, 4.94505053684115880965203501313, 6.72777861217198911545361132938, 7.50032305085772554762762688048, 8.241775652373091175400604039599, 9.395465568665968564030510328647, 9.812276396599836246537047025781, 10.91672201656036037690625746521

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