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Properties

Label 2-3332-476.387-c0-0-2
Degree $2$
Conductor $3332$
Sign $0.704 + 0.710i$
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  + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (1.36 − 0.366i)5-s − 0.999i·8-s + (−0.866 − 0.5i)9-s + (−1.36 − 0.366i)10-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (0.499 + 0.866i)18-s + (0.999 + 0.999i)20-s + (0.866 − 0.5i)25-s + (1 + i)29-s + (0.866 − 0.499i)32-s − 0.999i·34-s − 0.999i·36-s + (1.36 − 0.366i)37-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (1.36 − 0.366i)5-s − 0.999i·8-s + (−0.866 − 0.5i)9-s + (−1.36 − 0.366i)10-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (0.499 + 0.866i)18-s + (0.999 + 0.999i)20-s + (0.866 − 0.5i)25-s + (1 + i)29-s + (0.866 − 0.499i)32-s − 0.999i·34-s − 0.999i·36-s + (1.36 − 0.366i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.704 + 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.704 + 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $0.704 + 0.710i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (863, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ 0.704 + 0.710i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.056746669\)
\(L(\frac12)\) \(\approx\) \(1.056746669\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 \)
17 \( 1 + (-0.5 - 0.866i)T \)
good3 \( 1 + (0.866 + 0.5i)T^{2} \)
5 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
11 \( 1 + (-0.866 - 0.5i)T^{2} \)
13 \( 1 + T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.866 - 0.5i)T^{2} \)
29 \( 1 + (-1 - i)T + iT^{2} \)
31 \( 1 + (0.866 + 0.5i)T^{2} \)
37 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
41 \( 1 + (-1 + i)T - iT^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 - iT^{2} \)
73 \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \)
79 \( 1 + (0.866 - 0.5i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (1 + i)T + iT^{2} \)
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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

−8.869025644282243413626102239528, −8.264956147773025671554994567182, −7.38619539968566350593813571599, −6.30005263831985852621532516100, −5.99452786644814829947380513637, −4.97161127033563395014800418316, −3.76177156672170888550589658377, −2.85335583804944230706555168065, −2.02057573513368101683399016705, −1.02106786007784933774402506917, 1.14770475034360088537504572959, 2.44091938312418767833749439081, 2.77339437860753291328789928088, 4.59377084417835667756505715140, 5.47919858997078262523898582176, 5.98688946444183971461784402312, 6.58299928367369410460163943906, 7.51952710425388980414753855115, 8.166724981433817450268869337216, 8.928848438306949665190837173232

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