Properties

Label 2-840-840.293-c0-0-1
Degree $2$
Conductor $840$
Sign $-0.229 - 0.973i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 0.707i)2-s + (−0.707 + 0.707i)3-s − 1.00i·4-s + (−0.707 − 0.707i)5-s − 1.00i·6-s + i·7-s + (0.707 + 0.707i)8-s − 1.00i·9-s + 1.00·10-s + 1.41·11-s + (0.707 + 0.707i)12-s + (−0.707 − 0.707i)14-s + 1.00·15-s − 1.00·16-s + (0.707 + 0.707i)18-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (−0.707 + 0.707i)3-s − 1.00i·4-s + (−0.707 − 0.707i)5-s − 1.00i·6-s + i·7-s + (0.707 + 0.707i)8-s − 1.00i·9-s + 1.00·10-s + 1.41·11-s + (0.707 + 0.707i)12-s + (−0.707 − 0.707i)14-s + 1.00·15-s − 1.00·16-s + (0.707 + 0.707i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.229 - 0.973i$
Analytic conductor: \(0.419214\)
Root analytic conductor: \(0.647467\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :0),\ -0.229 - 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4896236624\)
\(L(\frac12)\) \(\approx\) \(0.4896236624\)
\(L(1)\) 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 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (0.707 + 0.707i)T \)
7 \( 1 - iT \)
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 \( 1 + (-1 - i)T + iT^{2} \)
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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

−10.56036348914369909976436189427, −9.507376332513271597442946892600, −8.922597913545683929451104444908, −8.474954288370207451266114004223, −7.09625253732592273389670699645, −6.36791357826312588518274170699, −5.35456511869911798075528263888, −4.75427825159890925897174862724, −3.55255530132479921596922979544, −1.34108280495155758038607577854, 0.808146649981180763141789924352, 2.25844481267753838998040089743, 3.74190083982064994031321079220, 4.36144758484577729371650016159, 6.21183307845880297793203296716, 6.92531020987349327793986997360, 7.60830839848579287449144212332, 8.280410840573720331898018058079, 9.588533819693222175183673687480, 10.28575046308627946864190954219

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