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Properties

Label 2-3321-369.155-c0-0-3
Degree $2$
Conductor $3321$
Sign $0.715 + 0.698i$
Analytic cond. $1.65739$
Root an. cond. $1.28739$
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  + (0.5 + 0.866i)4-s + (−0.707 − 1.22i)5-s + (0.965 − 0.258i)11-s + (−1.36 − 0.366i)13-s + (−0.499 + 0.866i)16-s + (0.707 − 1.22i)20-s + (1.22 − 0.707i)23-s + (−0.499 + 0.866i)25-s + (0.965 − 0.258i)29-s + (−0.5 − 0.866i)31-s + 37-s + (0.258 − 0.965i)41-s + (0.866 + 0.5i)43-s + (0.707 + 0.707i)44-s + (−0.258 − 0.965i)47-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)4-s + (−0.707 − 1.22i)5-s + (0.965 − 0.258i)11-s + (−1.36 − 0.366i)13-s + (−0.499 + 0.866i)16-s + (0.707 − 1.22i)20-s + (1.22 − 0.707i)23-s + (−0.499 + 0.866i)25-s + (0.965 − 0.258i)29-s + (−0.5 − 0.866i)31-s + 37-s + (0.258 − 0.965i)41-s + (0.866 + 0.5i)43-s + (0.707 + 0.707i)44-s + (−0.258 − 0.965i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3321 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.715 + 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3321 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.715 + 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3321\)    =    \(3^{4} \cdot 41\)
Sign: $0.715 + 0.698i$
Analytic conductor: \(1.65739\)
Root analytic conductor: \(1.28739\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3321} (2861, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3321,\ (\ :0),\ 0.715 + 0.698i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.207071512\)
\(L(\frac12)\) \(\approx\) \(1.207071512\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
41 \( 1 + (-0.258 + 0.965i)T \)
good2 \( 1 + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (-0.866 + 0.5i)T^{2} \)
11 \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \)
13 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 - T + T^{2} \)
43 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \)
53 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
59 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.866 - 0.5i)T^{2} \)
71 \( 1 + iT^{2} \)
73 \( 1 - iT - T^{2} \)
79 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
83 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
97 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
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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

−8.750599209349670937499047778558, −7.942591812927317593152976426650, −7.38147485757619691257689319065, −6.66969469044254048817897009075, −5.64324040389738341620123232819, −4.61299811651158778039776625279, −4.20424374550751640020390618959, −3.17967024399311028056632901398, −2.24725803269322213558457160931, −0.789938348023363311444204478846, 1.30623527928869487464605292435, 2.53194773797242469502359887095, 3.16353072121433506183145249363, 4.32931680907284891001929993688, 5.04408873232377515170461701974, 6.12679248685908949593121727703, 6.80012307560789370471213237416, 7.19885589413905183448867778843, 7.85981909703880064180228551018, 9.272996122638364964141169911637

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