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Properties

Label 2-333-37.36-c5-0-76
Degree $2$
Conductor $333$
Sign $0.938 - 0.344i$
Analytic cond. $53.4078$
Root an. cond. $7.30806$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8.11i·2-s − 33.8·4-s − 43.8i·5-s − 152.·7-s + 15.1i·8-s − 356.·10-s − 531.·11-s − 638. i·13-s + 1.23e3i·14-s − 960.·16-s − 1.62e3i·17-s + 860. i·19-s + 1.48e3i·20-s + 4.31e3i·22-s − 2.50e3i·23-s + ⋯
L(s)  = 1  − 1.43i·2-s − 1.05·4-s − 0.784i·5-s − 1.17·7-s + 0.0834i·8-s − 1.12·10-s − 1.32·11-s − 1.04i·13-s + 1.68i·14-s − 0.938·16-s − 1.36i·17-s + 0.546i·19-s + 0.830i·20-s + 1.89i·22-s − 0.987i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 333 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 - 0.344i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 333 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.938 - 0.344i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(333\)    =    \(3^{2} \cdot 37\)
Sign: $0.938 - 0.344i$
Analytic conductor: \(53.4078\)
Root analytic conductor: \(7.30806\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{333} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 333,\ (\ :5/2),\ 0.938 - 0.344i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.3256212191\)
\(L(\frac12)\) \(\approx\) \(0.3256212191\)
\(L(\frac{7}{2})\) 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 \( 1 + (-7.81e3 + 2.86e3i)T \)
good2 \( 1 + 8.11iT - 32T^{2} \)
5 \( 1 + 43.8iT - 3.12e3T^{2} \)
7 \( 1 + 152.T + 1.68e4T^{2} \)
11 \( 1 + 531.T + 1.61e5T^{2} \)
13 \( 1 + 638. iT - 3.71e5T^{2} \)
17 \( 1 + 1.62e3iT - 1.41e6T^{2} \)
19 \( 1 - 860. iT - 2.47e6T^{2} \)
23 \( 1 + 2.50e3iT - 6.43e6T^{2} \)
29 \( 1 + 2.32e3iT - 2.05e7T^{2} \)
31 \( 1 - 4.84e3iT - 2.86e7T^{2} \)
41 \( 1 + 1.62e4T + 1.15e8T^{2} \)
43 \( 1 - 6.23e3iT - 1.47e8T^{2} \)
47 \( 1 - 6.21e3T + 2.29e8T^{2} \)
53 \( 1 - 2.25e4T + 4.18e8T^{2} \)
59 \( 1 - 4.97e4iT - 7.14e8T^{2} \)
61 \( 1 - 5.25e3iT - 8.44e8T^{2} \)
67 \( 1 - 2.25e4T + 1.35e9T^{2} \)
71 \( 1 + 2.13e4T + 1.80e9T^{2} \)
73 \( 1 - 6.55e3T + 2.07e9T^{2} \)
79 \( 1 + 1.78e4iT - 3.07e9T^{2} \)
83 \( 1 - 2.96e4T + 3.93e9T^{2} \)
89 \( 1 + 1.56e4iT - 5.58e9T^{2} \)
97 \( 1 - 9.22e4iT - 8.58e9T^{2} \)
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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

−10.13533836755560732180782371601, −9.272997469095201159381178541054, −8.279038015778971854195044906939, −7.00031485272093716375855529972, −5.57420548536356385069516718114, −4.55873221968192741997259573418, −3.16875649462337369458377435957, −2.56115238474808305276148425625, −0.860945563032205476227190685085, −0.10617945246424396980785674479, 2.33778413986478369114994270104, 3.62521521542182467334202851231, 5.07290342515027025779899067215, 6.14595059463041421003449042392, 6.74980881533932422950823220371, 7.55811761687495959082337671801, 8.555559569484829791959628671160, 9.590896917563504266115465169218, 10.52874635197072631102894403776, 11.48980518712323448083084298479

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