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Properties

Label 2-3332-3332.2311-c0-0-2
Degree $2$
Conductor $3332$
Sign $0.518 - 0.855i$
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  + (0.623 + 0.781i)2-s + (1.62 − 0.781i)3-s + (−0.222 + 0.974i)4-s + (1.62 + 0.781i)6-s + (−0.900 + 0.433i)7-s + (−0.900 + 0.433i)8-s + (1.40 − 1.75i)9-s + (0.777 + 0.974i)11-s + (0.400 + 1.75i)12-s + (0.777 + 0.974i)13-s + (−0.900 − 0.433i)14-s + (−0.900 − 0.433i)16-s + (−0.222 − 0.974i)17-s + 2.24·18-s + (−1.12 + 1.40i)21-s + (−0.277 + 1.21i)22-s + ⋯
L(s)  = 1  + (0.623 + 0.781i)2-s + (1.62 − 0.781i)3-s + (−0.222 + 0.974i)4-s + (1.62 + 0.781i)6-s + (−0.900 + 0.433i)7-s + (−0.900 + 0.433i)8-s + (1.40 − 1.75i)9-s + (0.777 + 0.974i)11-s + (0.400 + 1.75i)12-s + (0.777 + 0.974i)13-s + (−0.900 − 0.433i)14-s + (−0.900 − 0.433i)16-s + (−0.222 − 0.974i)17-s + 2.24·18-s + (−1.12 + 1.40i)21-s + (−0.277 + 1.21i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.518 - 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.518 - 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $0.518 - 0.855i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (2311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ 0.518 - 0.855i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.794472636\)
\(L(\frac12)\) \(\approx\) \(2.794472636\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.623 - 0.781i)T \)
7 \( 1 + (0.900 - 0.433i)T \)
17 \( 1 + (0.222 + 0.974i)T \)
good3 \( 1 + (-1.62 + 0.781i)T + (0.623 - 0.781i)T^{2} \)
5 \( 1 + (-0.623 + 0.781i)T^{2} \)
11 \( 1 + (-0.777 - 0.974i)T + (-0.222 + 0.974i)T^{2} \)
13 \( 1 + (-0.777 - 0.974i)T + (-0.222 + 0.974i)T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.277 - 1.21i)T + (-0.900 - 0.433i)T^{2} \)
29 \( 1 + (0.900 - 0.433i)T^{2} \)
31 \( 1 + 1.80T + T^{2} \)
37 \( 1 + (0.900 - 0.433i)T^{2} \)
41 \( 1 + (-0.623 + 0.781i)T^{2} \)
43 \( 1 + (-0.623 - 0.781i)T^{2} \)
47 \( 1 + (0.222 - 0.974i)T^{2} \)
53 \( 1 + (-0.400 + 1.75i)T + (-0.900 - 0.433i)T^{2} \)
59 \( 1 + (-0.623 - 0.781i)T^{2} \)
61 \( 1 + (0.900 - 0.433i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.400 + 1.75i)T + (-0.900 - 0.433i)T^{2} \)
73 \( 1 + (0.222 + 0.974i)T^{2} \)
79 \( 1 - 1.24T + T^{2} \)
83 \( 1 + (0.222 + 0.974i)T^{2} \)
89 \( 1 + (1.12 - 1.40i)T + (-0.222 - 0.974i)T^{2} \)
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

−8.969187130111730971453306582358, −8.073406986954312727797597961172, −7.27780830565181572790483934944, −6.81375933264209262423243254519, −6.33066190217200478284695899443, −5.12526868749170219000549225796, −3.91918708090996316887330666189, −3.59836164524781691976066744660, −2.57563752139687124372397925501, −1.78167761231273215605079634324, 1.29223192304533435226780931431, 2.54755603560408072253024734036, 3.30978354497273307790501486054, 3.73745350884794681555926611202, 4.29502192060727304995159696964, 5.53402952251152962625524970097, 6.26884828633074220788346536417, 7.24447633704381236107552869307, 8.385251299017871601771258033438, 8.816847783384506408862439114889

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