↑

Properties

Label 2-3360-840.797-c0-0-10
Degree $2$
Conductor $3360$
Sign $-0.229 + 0.973i$
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  + (−0.707 − 0.707i)3-s + (0.707 − 0.707i)5-s + 7-s + 1.00i·9-s − 1.41·11-s − 1.00·15-s + (−0.707 − 0.707i)21-s − 1.00i·25-s + (0.707 − 0.707i)27-s − 1.41i·29-s − 2i·31-s + (1.00 + 1.00i)33-s + (0.707 − 0.707i)35-s + (0.707 + 0.707i)45-s + 49-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)3-s + (0.707 − 0.707i)5-s + 7-s + 1.00i·9-s − 1.41·11-s − 1.00·15-s + (−0.707 − 0.707i)21-s − 1.00i·25-s + (0.707 − 0.707i)27-s − 1.41i·29-s − 2i·31-s + (1.00 + 1.00i)33-s + (0.707 − 0.707i)35-s + (0.707 + 0.707i)45-s + 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3360\)    =    \(2^{5} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.229 + 0.973i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3360} (2897, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3360,\ (\ :0),\ -0.229 + 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.072206507\)
\(L(\frac12)\) \(\approx\) \(1.072206507\)
\(L(1)\) 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 \( 1 + (-0.707 + 0.707i)T \)
7 \( 1 - T \)
good11 \( 1 + 1.41T + T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + 1.41iT - T^{2} \)
31 \( 1 + 2iT - T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 - 1.41T + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (1 + i)T + iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (1 - i)T - iT^{2} \)
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

−8.216049902046553089152368692111, −7.989851831097416505859762794504, −7.20333208870545555022205527603, −6.05781391937905996396494269251, −5.63868314630088957736969817426, −4.93316276681664265789389373808, −4.27448996492997754182826278394, −2.48006486479540814297196404105, −1.97447066623838541783359502470, −0.71425747431589861088705840492, 1.46198703715848259801516380414, 2.65255869716228727201989170998, 3.48058843131609695830604290793, 4.69338518132098741202540077866, 5.25172465915438293034445005208, 5.73871818715661879794652564928, 6.80239376771649808066518204739, 7.33883979179595479860129468274, 8.413990533093743169958261670883, 8.994915895508182652286987550188

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