Properties

Label 2-3360-105.74-c0-0-5
Degree $2$
Conductor $3360$
Sign $-0.553 + 0.832i$
Analytic cond. $1.67685$
Root an. cond. $1.29493$
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 − 0.707i)3-s + (−0.866 + 0.5i)5-s + (−0.965 + 0.258i)7-s − 1.00i·9-s + (−0.258 + 0.965i)15-s + (−0.500 + 0.866i)21-s + (−0.965 − 1.67i)23-s + (0.499 − 0.866i)25-s + (−0.707 − 0.707i)27-s − 1.73i·29-s + (0.707 − 0.707i)35-s i·41-s + 0.517i·43-s + (0.500 + 0.866i)45-s + (−0.707 − 1.22i)47-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)3-s + (−0.866 + 0.5i)5-s + (−0.965 + 0.258i)7-s − 1.00i·9-s + (−0.258 + 0.965i)15-s + (−0.500 + 0.866i)21-s + (−0.965 − 1.67i)23-s + (0.499 − 0.866i)25-s + (−0.707 − 0.707i)27-s − 1.73i·29-s + (0.707 − 0.707i)35-s i·41-s + 0.517i·43-s + (0.500 + 0.866i)45-s + (−0.707 − 1.22i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3360\)    =    \(2^{5} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.553 + 0.832i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3360} (3329, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3360,\ (\ :0),\ -0.553 + 0.832i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8063279850\)
\(L(\frac12)\) \(\approx\) \(0.8063279850\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (0.965 - 0.258i)T \)
good11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.965 + 1.67i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + 1.73iT - T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + iT - T^{2} \)
43 \( 1 - 0.517iT - T^{2} \)
47 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (1.67 + 0.965i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + 0.517T + T^{2} \)
89 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−8.414942771597592131934409126378, −7.85668831776246140117252847465, −7.05273382291059803991542005594, −6.47936477786197127806356128521, −5.85626575723916940549114981350, −4.35797681119336000726478448259, −3.73616002315054518555519974145, −2.84223756950420830526606167491, −2.20379385791908296630401422755, −0.42669062067821938387986788612, 1.56159748332681379007208729406, 3.04911743384282214052360479200, 3.50958670979317578234957647762, 4.27766719934474511999680325249, 5.07535155254076354645552295172, 5.95350654470036380253677484460, 7.10124359835544428613973377741, 7.62542983614596970857715653170, 8.404653800292038519045282621695, 9.060640834086873965634731250805

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