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Properties

Label 2-3321-369.122-c0-0-5
Degree $2$
Conductor $3321$
Sign $-0.342 + 0.939i$
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  + (1.22 − 0.707i)2-s + (0.499 − 0.866i)4-s + (−1.22 − 0.707i)5-s + (1.22 − 0.707i)7-s − 2·10-s + (−0.5 − 0.866i)11-s + (1.22 + 0.707i)13-s + (0.999 − 1.73i)14-s + (0.499 + 0.866i)16-s − 1.41i·19-s + (−1.22 + 0.707i)20-s + (−1.22 − 0.707i)22-s + (0.499 + 0.866i)25-s + 2·26-s − 1.41i·28-s + (−0.5 − 0.866i)29-s + ⋯
L(s)  = 1  + (1.22 − 0.707i)2-s + (0.499 − 0.866i)4-s + (−1.22 − 0.707i)5-s + (1.22 − 0.707i)7-s − 2·10-s + (−0.5 − 0.866i)11-s + (1.22 + 0.707i)13-s + (0.999 − 1.73i)14-s + (0.499 + 0.866i)16-s − 1.41i·19-s + (−1.22 + 0.707i)20-s + (−1.22 − 0.707i)22-s + (0.499 + 0.866i)25-s + 2·26-s − 1.41i·28-s + (−0.5 − 0.866i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3321 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3321 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3321\)    =    \(3^{4} \cdot 41\)
Sign: $-0.342 + 0.939i$
Analytic conductor: \(1.65739\)
Root analytic conductor: \(1.28739\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3321} (1106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3321,\ (\ :0),\ -0.342 + 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.200136036\)
\(L(\frac12)\) \(\approx\) \(2.200136036\)
\(L(1)\) 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 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + 1.41iT - T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + T + T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + T + T^{2} \)
59 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 - T + T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−8.548236990922838297814727308183, −7.916750618144191565893814884616, −7.15940252420447541659590376727, −6.02595624003774170435724167841, −5.07235650501141816016986394624, −4.64371277624337885817031567538, −3.88794504998646651504360934915, −3.41085126964286305351964221216, −2.07785072908047716199889230414, −0.960692387728654989330389905862, 1.71285680744513568209347413170, 3.07182232501787303163258631968, 3.74915170504257816436857484215, 4.44180985963728711484148480847, 5.30189946057608921619104154211, 5.81625860231970661919462668032, 6.74765977157020201729349163740, 7.58303371793779048517206946914, 7.933974450817365008407502947552, 8.593686089410493039490122514751

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