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Properties

Label 2-3332-3332.3263-c0-0-2
Degree $2$
Conductor $3332$
Sign $0.926 + 0.375i$
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  + (−0.900 − 0.433i)2-s + (0.0990 + 0.433i)3-s + (0.623 + 0.781i)4-s + (0.0990 − 0.433i)6-s + (−0.222 − 0.974i)7-s + (−0.222 − 0.974i)8-s + (0.722 − 0.347i)9-s + (1.62 + 0.781i)11-s + (−0.277 + 0.347i)12-s + (1.62 + 0.781i)13-s + (−0.222 + 0.974i)14-s + (−0.222 + 0.974i)16-s + (0.623 − 0.781i)17-s − 0.801·18-s + (0.400 − 0.193i)21-s + (−1.12 − 1.40i)22-s + ⋯
L(s)  = 1  + (−0.900 − 0.433i)2-s + (0.0990 + 0.433i)3-s + (0.623 + 0.781i)4-s + (0.0990 − 0.433i)6-s + (−0.222 − 0.974i)7-s + (−0.222 − 0.974i)8-s + (0.722 − 0.347i)9-s + (1.62 + 0.781i)11-s + (−0.277 + 0.347i)12-s + (1.62 + 0.781i)13-s + (−0.222 + 0.974i)14-s + (−0.222 + 0.974i)16-s + (0.623 − 0.781i)17-s − 0.801·18-s + (0.400 − 0.193i)21-s + (−1.12 − 1.40i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 + 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 + 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $0.926 + 0.375i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (3263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ 0.926 + 0.375i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.074677317\)
\(L(\frac12)\) \(\approx\) \(1.074677317\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.900 + 0.433i)T \)
7 \( 1 + (0.222 + 0.974i)T \)
17 \( 1 + (-0.623 + 0.781i)T \)
good3 \( 1 + (-0.0990 - 0.433i)T + (-0.900 + 0.433i)T^{2} \)
5 \( 1 + (0.900 - 0.433i)T^{2} \)
11 \( 1 + (-1.62 - 0.781i)T + (0.623 + 0.781i)T^{2} \)
13 \( 1 + (-1.62 - 0.781i)T + (0.623 + 0.781i)T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (1.12 + 1.40i)T + (-0.222 + 0.974i)T^{2} \)
29 \( 1 + (0.222 + 0.974i)T^{2} \)
31 \( 1 + 0.445T + T^{2} \)
37 \( 1 + (0.222 + 0.974i)T^{2} \)
41 \( 1 + (0.900 - 0.433i)T^{2} \)
43 \( 1 + (0.900 + 0.433i)T^{2} \)
47 \( 1 + (-0.623 - 0.781i)T^{2} \)
53 \( 1 + (0.277 + 0.347i)T + (-0.222 + 0.974i)T^{2} \)
59 \( 1 + (0.900 + 0.433i)T^{2} \)
61 \( 1 + (0.222 + 0.974i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.277 + 0.347i)T + (-0.222 + 0.974i)T^{2} \)
73 \( 1 + (-0.623 + 0.781i)T^{2} \)
79 \( 1 + 1.80T + T^{2} \)
83 \( 1 + (-0.623 + 0.781i)T^{2} \)
89 \( 1 + (-0.400 + 0.193i)T + (0.623 - 0.781i)T^{2} \)
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

−9.064367908270269060946305347559, −8.152707662710964097548883423395, −7.26179947740138308019505865352, −6.70952275722340195837608770161, −6.16430894839717575852009848330, −4.38062174021191912654306776561, −3.99028868104225092765684075007, −3.41001227877780259508319474166, −1.82250609616824249987916364299, −1.10940514060402664446714313459, 1.27612303701564556537178517596, 1.81553218313765028260279962571, 3.25706194755196163332829296173, 4.04689507810119786006619538020, 5.71536426371494847239119233101, 5.88694964728132425346472724850, 6.56127637021715190624354710696, 7.54044513459968227206675373740, 8.259520707283768019916818097614, 8.629506462096311933827393236719

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