Properties

Label 2-3360-840.293-c0-0-7
Degree $2$
Conductor $3360$
Sign $0.850 + 0.525i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.382 + 0.923i)3-s + (−0.923 − 0.382i)5-s + (0.707 − 0.707i)7-s + (−0.707 + 0.707i)9-s + (−0.541 − 0.541i)13-s i·15-s − 1.84i·19-s + (0.923 + 0.382i)21-s + (1 − i)23-s + (0.707 + 0.707i)25-s + (−0.923 − 0.382i)27-s + (−0.923 + 0.382i)35-s + (0.292 − 0.707i)39-s + (0.923 − 0.382i)45-s − 1.00i·49-s + ⋯
L(s)  = 1  + (0.382 + 0.923i)3-s + (−0.923 − 0.382i)5-s + (0.707 − 0.707i)7-s + (−0.707 + 0.707i)9-s + (−0.541 − 0.541i)13-s i·15-s − 1.84i·19-s + (0.923 + 0.382i)21-s + (1 − i)23-s + (0.707 + 0.707i)25-s + (−0.923 − 0.382i)27-s + (−0.923 + 0.382i)35-s + (0.292 − 0.707i)39-s + (0.923 − 0.382i)45-s − 1.00i·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3360\)    =    \(2^{5} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.850 + 0.525i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3360} (1553, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3360,\ (\ :0),\ 0.850 + 0.525i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.154459525\)
\(L(\frac12)\) \(\approx\) \(1.154459525\)
\(L(1)\) 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 \( 1 + (-0.382 - 0.923i)T \)
5 \( 1 + (0.923 + 0.382i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
good11 \( 1 + T^{2} \)
13 \( 1 + (0.541 + 0.541i)T + iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + 1.84iT - T^{2} \)
23 \( 1 + (-1 + i)T - iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 - 1.84T + T^{2} \)
61 \( 1 + 0.765T + T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + 1.41iT - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 - 1.41iT - T^{2} \)
83 \( 1 + (-1.30 + 1.30i)T - iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.766337277396928769802671986319, −8.062022227407298838175041595353, −7.43664674732840630658972111644, −6.70319470400455643415034433900, −5.20241506143048614740122080604, −4.83894813273920517120413078318, −4.20614666696370018470202407295, −3.29795402103987348238764876364, −2.44111770790349918237505979927, −0.70419419749827836186568341279, 1.37415554644104943027881642219, 2.30631115694318573261645016259, 3.24940346485489654429837971108, 4.05208492218932404416221390740, 5.15950855214477065241005726149, 5.93371577913785198420169899792, 6.84211449547595388459120570967, 7.49068191283867264546759958396, 8.028289322395749786300008215537, 8.629087272652292175704265081912

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