↑

Properties

Label 2-840-840.797-c0-0-8
Degree $2$
Conductor $840$
Sign $0.525 + 0.850i$
Analytic cond. $0.419214$
Root an. cond. $0.647467$
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)2-s + (0.382 − 0.923i)3-s − 1.00i·4-s + (0.923 − 0.382i)5-s + (0.382 + 0.923i)6-s + (−0.707 − 0.707i)7-s + (0.707 + 0.707i)8-s + (−0.707 − 0.707i)9-s + (−0.382 + 0.923i)10-s + (−0.923 − 0.382i)12-s + (0.541 − 0.541i)13-s + 1.00·14-s − i·15-s − 1.00·16-s + 18-s + 1.84i·19-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (0.382 − 0.923i)3-s − 1.00i·4-s + (0.923 − 0.382i)5-s + (0.382 + 0.923i)6-s + (−0.707 − 0.707i)7-s + (0.707 + 0.707i)8-s + (−0.707 − 0.707i)9-s + (−0.382 + 0.923i)10-s + (−0.923 − 0.382i)12-s + (0.541 − 0.541i)13-s + 1.00·14-s − i·15-s − 1.00·16-s + 18-s + 1.84i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.525 + 0.850i$
Analytic conductor: \(0.419214\)
Root analytic conductor: \(0.647467\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (797, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :0),\ 0.525 + 0.850i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8216245359\)
\(L(\frac12)\) \(\approx\) \(0.8216245359\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
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

−10.07608222229433326313101296559, −9.350936988342803394229250459330, −8.354759036095003135260933968989, −7.899681547836369427091871455141, −6.74201963139575996194757334546, −6.20369324672167700527070517314, −5.47302253703198275946499325660, −3.83333554044364463845150378234, −2.25177379898839137727051979619, −1.06855885526084252602384235126, 2.10125914649203257964519371285, 2.90021769796185125680223124274, 3.85802937388195239787538646251, 5.13671228873837023086124174410, 6.23421030000519011560793597046, 7.22568927876219950228783751460, 8.557405598661689586937062586312, 9.091477230409969849992775866667, 9.693994625341326937804271021509, 10.27851602629375837483109042858

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