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Properties

Label 2-3332-3332.1563-c0-0-3
Degree $2$
Conductor $3332$
Sign $0.999 + 0.0213i$
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  + (0.988 − 0.149i)2-s + (0.930 + 0.634i)3-s + (0.955 − 0.294i)4-s + (1.01 + 0.488i)6-s + (−0.433 − 0.900i)7-s + (0.900 − 0.433i)8-s + (0.0983 + 0.250i)9-s + (0.108 − 0.277i)11-s + (1.07 + 0.332i)12-s + (1.23 + 1.54i)13-s + (−0.563 − 0.826i)14-s + (0.826 − 0.563i)16-s + (−0.733 + 0.680i)17-s + (0.134 + 0.233i)18-s + (0.167 − 1.11i)21-s + (0.0663 − 0.290i)22-s + ⋯
L(s)  = 1  + (0.988 − 0.149i)2-s + (0.930 + 0.634i)3-s + (0.955 − 0.294i)4-s + (1.01 + 0.488i)6-s + (−0.433 − 0.900i)7-s + (0.900 − 0.433i)8-s + (0.0983 + 0.250i)9-s + (0.108 − 0.277i)11-s + (1.07 + 0.332i)12-s + (1.23 + 1.54i)13-s + (−0.563 − 0.826i)14-s + (0.826 − 0.563i)16-s + (−0.733 + 0.680i)17-s + (0.134 + 0.233i)18-s + (0.167 − 1.11i)21-s + (0.0663 − 0.290i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0213i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0213i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $0.999 + 0.0213i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (1563, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ 0.999 + 0.0213i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.172945597\)
\(L(\frac12)\) \(\approx\) \(3.172945597\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.988 + 0.149i)T \)
7 \( 1 + (0.433 + 0.900i)T \)
17 \( 1 + (0.733 - 0.680i)T \)
good3 \( 1 + (-0.930 - 0.634i)T + (0.365 + 0.930i)T^{2} \)
5 \( 1 + (0.988 + 0.149i)T^{2} \)
11 \( 1 + (-0.108 + 0.277i)T + (-0.733 - 0.680i)T^{2} \)
13 \( 1 + (-1.23 - 1.54i)T + (-0.222 + 0.974i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (1.14 + 1.06i)T + (0.0747 + 0.997i)T^{2} \)
29 \( 1 + (0.900 - 0.433i)T^{2} \)
31 \( 1 + (-0.433 - 0.751i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.826 - 0.563i)T^{2} \)
41 \( 1 + (-0.623 + 0.781i)T^{2} \)
43 \( 1 + (-0.623 - 0.781i)T^{2} \)
47 \( 1 + (-0.955 + 0.294i)T^{2} \)
53 \( 1 + (-0.142 + 0.0440i)T + (0.826 - 0.563i)T^{2} \)
59 \( 1 + (0.988 - 0.149i)T^{2} \)
61 \( 1 + (-0.826 - 0.563i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.443 - 1.94i)T + (-0.900 - 0.433i)T^{2} \)
73 \( 1 + (-0.955 - 0.294i)T^{2} \)
79 \( 1 + (0.930 - 1.61i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.222 + 0.974i)T^{2} \)
89 \( 1 + (0.603 + 1.53i)T + (-0.733 + 0.680i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−8.680205750723798275374108431520, −8.258702539342601167823860307736, −7.06821966396443575303997739997, −6.44637097173549866712401321398, −5.92092837026162417709018000788, −4.37872097165212621588311257686, −4.13532793319556336034403937233, −3.58041343158299684792528390613, −2.56171675389418960951696395805, −1.54004002385790361494389007272, 1.68348336970180104727869907784, 2.49820810780931327865075149239, 3.20725692565892619034164020378, 3.91197369453991451536463021001, 5.14076079650558823086126198496, 5.85911089089225862502525367822, 6.36958232782710139504348168194, 7.45782328997739615602591859800, 7.909879368965959248475930155761, 8.544093523673375429437388833369

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