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Properties

Label 2-3332-476.367-c0-0-3
Degree $2$
Conductor $3332$
Sign $0.695 + 0.718i$
Analytic cond. $1.66288$
Root an. cond. $1.28952$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.991 − 0.130i)2-s + (0.965 − 0.258i)4-s + (0.835 − 0.732i)5-s + (0.923 − 0.382i)8-s + (−0.608 + 0.793i)9-s + (0.732 − 0.835i)10-s + (−1 − i)13-s + (0.866 − 0.5i)16-s + (0.793 − 0.608i)17-s + (−0.499 + 0.866i)18-s + (0.617 − 0.923i)20-s + (0.0306 − 0.232i)25-s + (−1.12 − 0.860i)26-s + (−0.216 − 1.08i)29-s + (0.793 − 0.608i)32-s + ⋯
L(s)  = 1  + (0.991 − 0.130i)2-s + (0.965 − 0.258i)4-s + (0.835 − 0.732i)5-s + (0.923 − 0.382i)8-s + (−0.608 + 0.793i)9-s + (0.732 − 0.835i)10-s + (−1 − i)13-s + (0.866 − 0.5i)16-s + (0.793 − 0.608i)17-s + (−0.499 + 0.866i)18-s + (0.617 − 0.923i)20-s + (0.0306 − 0.232i)25-s + (−1.12 − 0.860i)26-s + (−0.216 − 1.08i)29-s + (0.793 − 0.608i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.695 + 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.695 + 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $0.695 + 0.718i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (1795, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ 0.695 + 0.718i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.655765195\)
\(L(\frac12)\) \(\approx\) \(2.655765195\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.991 + 0.130i)T \)
7 \( 1 \)
17 \( 1 + (-0.793 + 0.608i)T \)
good3 \( 1 + (0.608 - 0.793i)T^{2} \)
5 \( 1 + (-0.835 + 0.732i)T + (0.130 - 0.991i)T^{2} \)
11 \( 1 + (-0.991 + 0.130i)T^{2} \)
13 \( 1 + (1 + i)T + iT^{2} \)
19 \( 1 + (0.965 - 0.258i)T^{2} \)
23 \( 1 + (-0.608 - 0.793i)T^{2} \)
29 \( 1 + (0.216 + 1.08i)T + (-0.923 + 0.382i)T^{2} \)
31 \( 1 + (-0.608 + 0.793i)T^{2} \)
37 \( 1 + (0.108 - 1.65i)T + (-0.991 - 0.130i)T^{2} \)
41 \( 1 + (-0.0761 + 0.382i)T + (-0.923 - 0.382i)T^{2} \)
43 \( 1 + (0.707 - 0.707i)T^{2} \)
47 \( 1 + (0.866 + 0.5i)T^{2} \)
53 \( 1 + (-1.12 - 1.46i)T + (-0.258 + 0.965i)T^{2} \)
59 \( 1 + (-0.965 - 0.258i)T^{2} \)
61 \( 1 + (-0.630 - 1.85i)T + (-0.793 + 0.608i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.382 - 0.923i)T^{2} \)
73 \( 1 + (1.57 + 0.534i)T + (0.793 + 0.608i)T^{2} \)
79 \( 1 + (0.608 + 0.793i)T^{2} \)
83 \( 1 + (0.707 + 0.707i)T^{2} \)
89 \( 1 + (0.866 + 0.5i)T^{2} \)
97 \( 1 + (1.63 - 0.324i)T + (0.923 - 0.382i)T^{2} \)
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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

−8.643398774171405332026871177877, −7.79340509341281811360521437594, −7.25048524690780573808427788860, −6.07948821270604965305855747785, −5.45136593535720083868508903079, −5.12181641146545939122589498297, −4.23831885762843513765121234881, −2.93743532467631561619264879415, −2.44690739075334442720094291530, −1.24826894549178629095598830741, 1.75401684186257254299049495232, 2.56485190531222502261392815147, 3.39936150219773583379569857782, 4.18773484200273489544210493376, 5.32271259723078474361568681366, 5.77851115483670475186221545178, 6.69406988653687654377977071573, 6.96931963080234867251506297859, 7.972257529097417686756458556082, 8.946477700741616817725971939327

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