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Properties

Label 2-3332-3332.1359-c0-0-0
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: $\chi_{3332} (1359, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ 0.801 - 0.598i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02027737937\)
\(L(\frac12)\) \(\approx\) \(0.02027737937\)
\(L(1)\) 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

−9.201987860855850727088234395129, −8.066920187603464492786905126398, −7.36374700017835706605293277814, −6.74944744496767880065042927278, −5.92610897833473664406805713919, −5.04741072961767756225213803752, −4.01935910638189237534916565990, −3.31969567680309449118775789301, −2.11087475669093453480585777922, −1.17423292886629159964779486587, 0.01584294294396166476534273163, 2.10306961756013804235488997560, 3.55658383884694884193878397880, 4.37836332818426275165427131530, 5.05168114126877053188628448943, 5.79683370572585280460026038578, 6.27322886620008986522134259138, 7.09385886436738788531538900209, 8.089258567430141139942448924998, 8.716928596785286277723764021564

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