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Properties

Label 2-3360-105.44-c0-0-0
Degree $2$
Conductor $3360$
Sign $-0.0633 - 0.997i$
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  + (−0.965 + 0.258i)3-s + (−0.866 − 0.5i)5-s + (−0.258 + 0.965i)7-s + (0.866 − 0.499i)9-s + (0.965 + 0.258i)15-s − i·21-s + (0.258 − 0.448i)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 + 1.93i·43-s − 45-s + (−0.707 + 1.22i)47-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)3-s + (−0.866 − 0.5i)5-s + (−0.258 + 0.965i)7-s + (0.866 − 0.499i)9-s + (0.965 + 0.258i)15-s − i·21-s + (0.258 − 0.448i)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 + 1.93i·43-s − 45-s + (−0.707 + 1.22i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3360\)    =    \(2^{5} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.0633 - 0.997i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3360} (1409, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3360,\ (\ :0),\ -0.0633 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5282076591\)
\(L(\frac12)\) \(\approx\) \(0.5282076591\)
\(L(1)\) 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 \( 1 + (0.965 - 0.258i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (0.258 - 0.965i)T \)
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 \( 1 + (-0.258 + 0.448i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 - 1.93iT - T^{2} \)
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 \( 1 + (-0.448 + 0.258i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - 1.93T + T^{2} \)
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

−9.172189863027620483295488421134, −8.128225169620690005434492580106, −7.64945071771355307949971915025, −6.44188088405126942790818291407, −6.10907006908632004938003840101, −5.05154207822042076312803723292, −4.58144835350433611549272172053, −3.67468599679942541534562726457, −2.60234222677032192678699537807, −1.11800789827827589191487110764, 0.42420924918273687271557559479, 1.76384034667518348539089468111, 3.28087587167552501671153106500, 3.90001133185999087651566928571, 4.81241458089678872255578060959, 5.54318744237412182128772267587, 6.69915725865853952294020853192, 6.96825607112059202061029299741, 7.60585667502200229751105431485, 8.427742701283115090504833289016

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