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Properties

Label 2-3332-476.291-c0-0-3
Degree $2$
Conductor $3332$
Sign $-0.972 - 0.233i$
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  + (0.258 − 0.965i)2-s + (−0.866 − 0.499i)4-s + (1.12 − 1.46i)5-s + (−0.707 + 0.707i)8-s + (−0.965 − 0.258i)9-s + (−1.12 − 1.46i)10-s + (0.500 + 0.866i)16-s + (−0.965 + 0.258i)17-s + (−0.499 + 0.866i)18-s + (−1.70 + 0.707i)20-s + (−0.624 − 2.33i)25-s + (−0.707 − 1.70i)29-s + (0.965 − 0.258i)32-s + i·34-s + (0.707 + 0.707i)36-s + (−1.46 − 1.12i)37-s + ⋯
L(s)  = 1  + (0.258 − 0.965i)2-s + (−0.866 − 0.499i)4-s + (1.12 − 1.46i)5-s + (−0.707 + 0.707i)8-s + (−0.965 − 0.258i)9-s + (−1.12 − 1.46i)10-s + (0.500 + 0.866i)16-s + (−0.965 + 0.258i)17-s + (−0.499 + 0.866i)18-s + (−1.70 + 0.707i)20-s + (−0.624 − 2.33i)25-s + (−0.707 − 1.70i)29-s + (0.965 − 0.258i)32-s + i·34-s + (0.707 + 0.707i)36-s + (−1.46 − 1.12i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 - 0.233i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 - 0.233i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $-0.972 - 0.233i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (1243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ -0.972 - 0.233i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.142440303\)
\(L(\frac12)\) \(\approx\) \(1.142440303\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 + 0.965i)T \)
7 \( 1 \)
17 \( 1 + (0.965 - 0.258i)T \)
good3 \( 1 + (0.965 + 0.258i)T^{2} \)
5 \( 1 + (-1.12 + 1.46i)T + (-0.258 - 0.965i)T^{2} \)
11 \( 1 + (-0.258 + 0.965i)T^{2} \)
13 \( 1 - T^{2} \)
19 \( 1 + (0.866 + 0.5i)T^{2} \)
23 \( 1 + (0.965 - 0.258i)T^{2} \)
29 \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \)
31 \( 1 + (0.965 + 0.258i)T^{2} \)
37 \( 1 + (1.46 + 1.12i)T + (0.258 + 0.965i)T^{2} \)
41 \( 1 + (0.292 - 0.707i)T + (-0.707 - 0.707i)T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
59 \( 1 + (0.866 - 0.5i)T^{2} \)
61 \( 1 + (-0.758 + 0.0999i)T + (0.965 - 0.258i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.707 + 0.707i)T^{2} \)
73 \( 1 + (-1.83 - 0.241i)T + (0.965 + 0.258i)T^{2} \)
79 \( 1 + (0.965 - 0.258i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.707 - 1.70i)T + (-0.707 + 0.707i)T^{2} \)
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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

−8.605806865736493017659059497601, −8.189322803013431255270351322054, −6.60694446302236595815407365992, −5.71905655997961533965453186555, −5.41002486286836278034573531181, −4.50990933550106358646456071087, −3.74648874306236670675154747446, −2.43011682657594202268918753013, −1.87578948984808099513278947038, −0.58539248629331278947778383125, 2.04036288934781870613153549600, 2.96917741610359589758718964647, 3.62955722604908126287703879846, 5.01896021867004421173504562248, 5.53836384695720570934189535091, 6.29800337338599488824301582589, 6.90454733560109414127536949830, 7.36326093813315675667726618535, 8.550003453549906757520847215807, 8.968604085769015501170310569408

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