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Properties

Label 2-3332-3332.407-c0-0-2
Degree $2$
Conductor $3332$
Sign $0.801 + 0.598i$
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  + (−0.222 + 0.974i)2-s + (0.777 − 0.974i)3-s + (−0.900 − 0.433i)4-s + (0.777 + 0.974i)6-s + (0.623 − 0.781i)7-s + (0.623 − 0.781i)8-s + (−0.123 − 0.541i)9-s + (0.0990 − 0.433i)11-s + (−1.12 + 0.541i)12-s + (0.0990 − 0.433i)13-s + (0.623 + 0.781i)14-s + (0.623 + 0.781i)16-s + (−0.900 + 0.433i)17-s + 0.554·18-s + (−0.277 − 1.21i)21-s + (0.400 + 0.193i)22-s + ⋯
L(s)  = 1  + (−0.222 + 0.974i)2-s + (0.777 − 0.974i)3-s + (−0.900 − 0.433i)4-s + (0.777 + 0.974i)6-s + (0.623 − 0.781i)7-s + (0.623 − 0.781i)8-s + (−0.123 − 0.541i)9-s + (0.0990 − 0.433i)11-s + (−1.12 + 0.541i)12-s + (0.0990 − 0.433i)13-s + (0.623 + 0.781i)14-s + (0.623 + 0.781i)16-s + (−0.900 + 0.433i)17-s + 0.554·18-s + (−0.277 − 1.21i)21-s + (0.400 + 0.193i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 + 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 + 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $0.801 + 0.598i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (407, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ 0.801 + 0.598i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.415068823\)
\(L(\frac12)\) \(\approx\) \(1.415068823\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.222 - 0.974i)T \)
7 \( 1 + (-0.623 + 0.781i)T \)
17 \( 1 + (0.900 - 0.433i)T \)
good3 \( 1 + (-0.777 + 0.974i)T + (-0.222 - 0.974i)T^{2} \)
5 \( 1 + (0.222 + 0.974i)T^{2} \)
11 \( 1 + (-0.0990 + 0.433i)T + (-0.900 - 0.433i)T^{2} \)
13 \( 1 + (-0.0990 + 0.433i)T + (-0.900 - 0.433i)T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-0.400 - 0.193i)T + (0.623 + 0.781i)T^{2} \)
29 \( 1 + (-0.623 + 0.781i)T^{2} \)
31 \( 1 - 1.24T + T^{2} \)
37 \( 1 + (-0.623 + 0.781i)T^{2} \)
41 \( 1 + (0.222 + 0.974i)T^{2} \)
43 \( 1 + (0.222 - 0.974i)T^{2} \)
47 \( 1 + (0.900 + 0.433i)T^{2} \)
53 \( 1 + (1.12 + 0.541i)T + (0.623 + 0.781i)T^{2} \)
59 \( 1 + (0.222 - 0.974i)T^{2} \)
61 \( 1 + (-0.623 + 0.781i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (1.12 + 0.541i)T + (0.623 + 0.781i)T^{2} \)
73 \( 1 + (0.900 - 0.433i)T^{2} \)
79 \( 1 + 0.445T + T^{2} \)
83 \( 1 + (0.900 - 0.433i)T^{2} \)
89 \( 1 + (0.277 + 1.21i)T + (-0.900 + 0.433i)T^{2} \)
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

−8.396255741131433372408236078824, −8.011981819867755251076319122526, −7.35896781344060731639351647068, −6.66054513027771580912661659015, −6.06284910266007748167306223539, −4.91679078249261271172406145374, −4.25594542847401194036956686213, −3.16864521892414047524088377917, −1.92063486695949818513799242389, −0.906870331352341292476389969740, 1.56155869353784588912339738545, 2.54178433631714906095271459887, 3.16384483852969260352581428576, 4.28736852378233598901268534655, 4.59501675266173305164309397265, 5.49912617461983661980282345266, 6.74578121952291483845094408159, 7.80405078711136000542316992611, 8.520954206152058642092261634863, 9.068828447965179267013278841147

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