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Properties

Label 2-3321-369.122-c0-0-1
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

\(L(\frac{1}{2})\) \(\approx\) \(0.8985737039\)
\(L(\frac12)\) \(\approx\) \(0.8985737039\)
\(L(1)\) 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

−8.849427393183451914577041380466, −8.327788873578296959904218636589, −7.34649015352303876994746378445, −6.63252234998424726968656526051, −5.49111603891626796404006821730, −5.07301525581621405783067919194, −4.15501082669332077976799715563, −3.52174956888394239947654207450, −2.85762694786984019068337747812, −1.70162416262980296909862500614, 0.34007879133708174705865282261, 2.75070031320866153680977417423, 3.37280996950789680439383996765, 4.08272458447687541169463226847, 4.60621404796361914048825980100, 5.73212759158099750880831595016, 6.57125316222275424541713155583, 7.07259657403219428639838212326, 7.33564730093729640856323058878, 8.438936820544163773402793667051

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