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Properties

Label 2-840-8.5-c1-0-42
Degree $2$
Conductor $840$
Sign $-0.669 - 0.742i$
Analytic cond. $6.70743$
Root an. cond. $2.58987$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.342 − 1.37i)2-s − i·3-s + (−1.76 + 0.939i)4-s + i·5-s + (−1.37 + 0.342i)6-s − 7-s + (1.89 + 2.10i)8-s − 9-s + (1.37 − 0.342i)10-s − 4.47i·11-s + (0.939 + 1.76i)12-s − 4.04i·13-s + (0.342 + 1.37i)14-s + 15-s + (2.23 − 3.31i)16-s + 0.406·17-s + ⋯
L(s)  = 1  + (−0.242 − 0.970i)2-s − 0.577i·3-s + (−0.882 + 0.469i)4-s + 0.447i·5-s + (−0.560 + 0.139i)6-s − 0.377·7-s + (0.669 + 0.742i)8-s − 0.333·9-s + (0.433 − 0.108i)10-s − 1.34i·11-s + (0.271 + 0.509i)12-s − 1.12i·13-s + (0.0914 + 0.366i)14-s + 0.258·15-s + (0.558 − 0.829i)16-s + 0.0986·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.669 - 0.742i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.669 - 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.669 - 0.742i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (421, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ -0.669 - 0.742i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.171618 + 0.385594i\)
\(L(\frac12)\) \(\approx\) \(0.171618 + 0.385594i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.342 + 1.37i)T \)
3 \( 1 + iT \)
5 \( 1 - iT \)
7 \( 1 + T \)
good11 \( 1 + 4.47iT - 11T^{2} \)
13 \( 1 + 4.04iT - 13T^{2} \)
17 \( 1 - 0.406T + 17T^{2} \)
19 \( 1 - 1.45iT - 19T^{2} \)
23 \( 1 + 7.46T + 23T^{2} \)
29 \( 1 - 10.3iT - 29T^{2} \)
31 \( 1 + 8.18T + 31T^{2} \)
37 \( 1 + 3.35iT - 37T^{2} \)
41 \( 1 + 8.85T + 41T^{2} \)
43 \( 1 - 0.733iT - 43T^{2} \)
47 \( 1 + 1.71T + 47T^{2} \)
53 \( 1 + 1.37iT - 53T^{2} \)
59 \( 1 + 8.61iT - 59T^{2} \)
61 \( 1 - 6.60iT - 61T^{2} \)
67 \( 1 + 14.5iT - 67T^{2} \)
71 \( 1 + 15.1T + 71T^{2} \)
73 \( 1 - 7.02T + 73T^{2} \)
79 \( 1 - 11.0T + 79T^{2} \)
83 \( 1 + 5.72iT - 83T^{2} \)
89 \( 1 + 12.6T + 89T^{2} \)
97 \( 1 + 3.29T + 97T^{2} \)
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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

−9.853869345197188651663543347290, −8.793242005486269806682847279301, −8.153012621977340413292700481265, −7.30085141322048181142531182102, −6.06180254524832233356077185044, −5.28767233094474261791017039812, −3.58733867289117908786107231692, −3.13229118540920223243706880721, −1.75817301203454028652427158608, −0.21893450936345267252537407537, 1.94430296340110374354680120554, 3.96365494741096106107410263702, 4.49399301358207876127956586158, 5.51610965161214976135398234898, 6.45516320454481528917164614050, 7.30252475987702915803376504352, 8.204669095673580684143811346021, 9.127416260390659239992644344294, 9.740504158285208025004810247947, 10.22055352700166213489835748090

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