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Properties

Label 2-3332-3332.407-c0-0-3
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 + (0.974 − 1.22i)3-s + (−0.900 − 0.433i)4-s + (−0.974 − 1.22i)6-s + (−0.781 − 0.623i)7-s + (−0.623 + 0.781i)8-s + (−0.321 − 1.40i)9-s + (0.433 − 1.90i)11-s + (−1.40 + 0.678i)12-s + (−0.0990 + 0.433i)13-s + (−0.781 + 0.623i)14-s + (0.623 + 0.781i)16-s + (−0.900 + 0.433i)17-s − 1.44·18-s + (−1.52 + 0.347i)21-s + (−1.75 − 0.846i)22-s + ⋯
L(s)  = 1  + (0.222 − 0.974i)2-s + (0.974 − 1.22i)3-s + (−0.900 − 0.433i)4-s + (−0.974 − 1.22i)6-s + (−0.781 − 0.623i)7-s + (−0.623 + 0.781i)8-s + (−0.321 − 1.40i)9-s + (0.433 − 1.90i)11-s + (−1.40 + 0.678i)12-s + (−0.0990 + 0.433i)13-s + (−0.781 + 0.623i)14-s + (0.623 + 0.781i)16-s + (−0.900 + 0.433i)17-s − 1.44·18-s + (−1.52 + 0.347i)21-s + (−1.75 − 0.846i)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

\(L(\frac{1}{2})\) \(\approx\) \(1.441968462\)
\(L(\frac12)\) \(\approx\) \(1.441968462\)
\(L(1)\) 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 \( 1 + (0.781 + 0.623i)T \)
17 \( 1 + (0.900 - 0.433i)T \)
good3 \( 1 + (-0.974 + 1.22i)T + (-0.222 - 0.974i)T^{2} \)
5 \( 1 + (0.222 + 0.974i)T^{2} \)
11 \( 1 + (-0.433 + 1.90i)T + (-0.900 - 0.433i)T^{2} \)
13 \( 1 + (0.0990 - 0.433i)T + (-0.900 - 0.433i)T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-1.75 - 0.846i)T + (0.623 + 0.781i)T^{2} \)
29 \( 1 + (-0.623 + 0.781i)T^{2} \)
31 \( 1 + 1.56T + T^{2} \)
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 \( 1 + (-1.40 - 0.678i)T + (0.623 + 0.781i)T^{2} \)
73 \( 1 + (0.900 - 0.433i)T^{2} \)
79 \( 1 - 1.94T + T^{2} \)
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

−8.524691492170903427233193785782, −7.79085357911255892002101448776, −6.75910726388546770122495843532, −6.33568483139966082694483107763, −5.29932559112299617042797391408, −3.95971669711100366207224469614, −3.40085986949927219624964267328, −2.73343573491181054499224866888, −1.68636441992371162168697999859, −0.71123287276723704805990679413, 2.27245219337206432788446173304, 3.19682426352975954174877618798, 3.89270618416553483477381123303, 4.84200406741456707675548547400, 5.09265598927836257715264861051, 6.38458859493957620044091823697, 7.07155081868826347285518930436, 7.70264140682374544988197389849, 8.809816852457762940966524483293, 9.207184294612230344918471480224

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