Properties

Label 2-3330-185.184-c1-0-81
Degree $2$
Conductor $3330$
Sign $-0.808 + 0.588i$
Analytic cond. $26.5901$
Root an. cond. $5.15656$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + (−2 + i)5-s + 3i·7-s + 8-s + (−2 + i)10-s + 3·11-s − 4·13-s + 3i·14-s + 16-s − 7·17-s − 4i·19-s + (−2 + i)20-s + 3·22-s − 6·23-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + (−0.894 + 0.447i)5-s + 1.13i·7-s + 0.353·8-s + (−0.632 + 0.316i)10-s + 0.904·11-s − 1.10·13-s + 0.801i·14-s + 0.250·16-s − 1.69·17-s − 0.917i·19-s + (−0.447 + 0.223i)20-s + 0.639·22-s − 1.25·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.808 + 0.588i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.808 + 0.588i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3330\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 37\)
Sign: $-0.808 + 0.588i$
Analytic conductor: \(26.5901\)
Root analytic conductor: \(5.15656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3330} (739, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 3330,\ (\ :1/2),\ -0.808 + 0.588i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
5 \( 1 + (2 - i)T \)
37 \( 1 + (-6 + i)T \)
good7 \( 1 - 3iT - 7T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 + 7T + 17T^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 + 6T + 23T^{2} \)
29 \( 1 - 9iT - 29T^{2} \)
31 \( 1 + 5iT - 31T^{2} \)
41 \( 1 + 7T + 41T^{2} \)
43 \( 1 - T + 43T^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 + iT - 53T^{2} \)
59 \( 1 + 6iT - 59T^{2} \)
61 \( 1 + 5iT - 61T^{2} \)
67 \( 1 + 2iT - 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + 4iT - 73T^{2} \)
79 \( 1 + 4iT - 79T^{2} \)
83 \( 1 - 14iT - 83T^{2} \)
89 \( 1 + 6iT - 89T^{2} \)
97 \( 1 - 7T + 97T^{2} \)
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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

−8.359868199944863199021240091301, −7.35996393949692343753206835153, −6.76706723812939217514339957131, −6.16840648165814891284467185500, −5.09850215650327575403620761412, −4.47838947328881990374318444354, −3.67240067800049302071045514792, −2.67003249263391527849169126892, −2.03598916193688812775897295309, 0, 1.38628598957560626554783562742, 2.58452729523013995160084059365, 3.85232890071935981437975915187, 4.21866124720168452169509188987, 4.74289753763430731219637328693, 5.96997046907787308616731441224, 6.69920253220864371515547146483, 7.41021361158273499063481624693, 7.956555329998575745283673494655

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