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Properties

Label 2-3321-369.32-c0-0-2
Degree $2$
Conductor $3321$
Sign $-0.0988 - 0.995i$
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  + (0.707 + 1.22i)2-s + (−0.499 + 0.866i)4-s + (−1.36 − 0.366i)7-s + (0.258 − 0.965i)11-s + (−0.517 − 1.93i)14-s + (0.499 + 0.866i)16-s + (1 + i)19-s + (1.36 − 0.366i)22-s + (1.22 + 0.707i)23-s + (0.5 + 0.866i)25-s + (0.999 − i)28-s + (−0.258 + 0.965i)29-s + (0.5 − 0.866i)31-s + (−0.707 + 1.22i)32-s + 37-s + (−0.517 + 1.93i)38-s + ⋯
L(s)  = 1  + (0.707 + 1.22i)2-s + (−0.499 + 0.866i)4-s + (−1.36 − 0.366i)7-s + (0.258 − 0.965i)11-s + (−0.517 − 1.93i)14-s + (0.499 + 0.866i)16-s + (1 + i)19-s + (1.36 − 0.366i)22-s + (1.22 + 0.707i)23-s + (0.5 + 0.866i)25-s + (0.999 − i)28-s + (−0.258 + 0.965i)29-s + (0.5 − 0.866i)31-s + (−0.707 + 1.22i)32-s + 37-s + (−0.517 + 1.93i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3321 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0988 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3321 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0988 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3321\)    =    \(3^{4} \cdot 41\)
Sign: $-0.0988 - 0.995i$
Analytic conductor: \(1.65739\)
Root analytic conductor: \(1.28739\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3321} (1754, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3321,\ (\ :0),\ -0.0988 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.744173675\)
\(L(\frac12)\) \(\approx\) \(1.744173675\)
\(L(1)\) 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 \( 1 + (0.965 - 0.258i)T \)
good2 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
11 \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \)
13 \( 1 + (-0.866 + 0.5i)T^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + (-1 - i)T + iT^{2} \)
23 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \)
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 \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \)
53 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
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 \( 1 + (0.866 + 0.5i)T^{2} \)
83 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
97 \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \)
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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

−8.933317566917536683406993565003, −7.959685363297052601862342099423, −7.26327271194548831701804668428, −6.75268691046029347626417486674, −5.92038791501290166666644810578, −5.55301727819817532030098142239, −4.54334869025432451030736747122, −3.46725489840756632146839094135, −3.20291099191747204427852963831, −1.24304144505523948872915789913, 1.00676357541877848475951559275, 2.45817800753335202910462043210, 2.85514548024923133324033583978, 3.74153722887932990648874116625, 4.64607370463474638679727827730, 5.20643549728135220129526211923, 6.42715110773972280686056425880, 6.88242165299681313044786336701, 7.83846142843386937097425447360, 8.997071194922909756965631665951

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