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Properties

Label 2-840-35.27-c1-0-9
Degree $2$
Conductor $840$
Sign $0.569 - 0.822i$
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  + (0.707 + 0.707i)3-s + (2.21 − 0.315i)5-s + (0.706 + 2.54i)7-s + 1.00i·9-s − 1.16·11-s + (1.47 + 1.47i)13-s + (1.78 + 1.34i)15-s + (1.17 − 1.17i)17-s + 2.09·19-s + (−1.30 + 2.30i)21-s + (−5.33 + 5.33i)23-s + (4.80 − 1.39i)25-s + (−0.707 + 0.707i)27-s + 1.55i·29-s − 2.66i·31-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (0.990 − 0.141i)5-s + (0.266 + 0.963i)7-s + 0.333i·9-s − 0.352·11-s + (0.409 + 0.409i)13-s + (0.461 + 0.346i)15-s + (0.285 − 0.285i)17-s + 0.481·19-s + (−0.284 + 0.502i)21-s + (−1.11 + 1.11i)23-s + (0.960 − 0.279i)25-s + (−0.136 + 0.136i)27-s + 0.288i·29-s − 0.479i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 - 0.822i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.569 - 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.569 - 0.822i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ 0.569 - 0.822i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.90318 + 0.997280i\)
\(L(\frac12)\) \(\approx\) \(1.90318 + 0.997280i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (-2.21 + 0.315i)T \)
7 \( 1 + (-0.706 - 2.54i)T \)
good11 \( 1 + 1.16T + 11T^{2} \)
13 \( 1 + (-1.47 - 1.47i)T + 13iT^{2} \)
17 \( 1 + (-1.17 + 1.17i)T - 17iT^{2} \)
19 \( 1 - 2.09T + 19T^{2} \)
23 \( 1 + (5.33 - 5.33i)T - 23iT^{2} \)
29 \( 1 - 1.55iT - 29T^{2} \)
31 \( 1 + 2.66iT - 31T^{2} \)
37 \( 1 + (-0.549 - 0.549i)T + 37iT^{2} \)
41 \( 1 + 8.59iT - 41T^{2} \)
43 \( 1 + (5.86 - 5.86i)T - 43iT^{2} \)
47 \( 1 + (-8.22 + 8.22i)T - 47iT^{2} \)
53 \( 1 + (-3.22 + 3.22i)T - 53iT^{2} \)
59 \( 1 - 3.54T + 59T^{2} \)
61 \( 1 + 4.08iT - 61T^{2} \)
67 \( 1 + (-3.06 - 3.06i)T + 67iT^{2} \)
71 \( 1 + 1.65T + 71T^{2} \)
73 \( 1 + (-6.90 - 6.90i)T + 73iT^{2} \)
79 \( 1 - 12.6iT - 79T^{2} \)
83 \( 1 + (-3.91 - 3.91i)T + 83iT^{2} \)
89 \( 1 + 11.8T + 89T^{2} \)
97 \( 1 + (-0.533 + 0.533i)T - 97iT^{2} \)
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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

−10.03563139068040828788192559487, −9.546689794826226442620921136307, −8.756611594837831810534174681501, −8.023022108218897683853251992986, −6.83594750730514824311476768536, −5.63436415733007152658840093479, −5.29604867315486288628196051718, −3.91673744047961688026705956681, −2.67586402291791580065527520738, −1.73527083293774308024880116539, 1.11281647576507877720356343008, 2.33886635142585323092130029792, 3.49476632864670597022846492212, 4.69414300676854721465445459299, 5.84199003149938619794463513774, 6.59829968945942211893452802141, 7.58430148841031391990054174690, 8.273688703774688057106884829173, 9.250932877495252463048897168986, 10.27790732654672979204536547163

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