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Properties

Label 2-840-56.27-c1-0-60
Degree $2$
Conductor $840$
Sign $-0.998 + 0.0505i$
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  + (0.818 − 1.15i)2-s + i·3-s + (−0.658 − 1.88i)4-s + 5-s + (1.15 + 0.818i)6-s + (−0.864 − 2.50i)7-s + (−2.71 − 0.787i)8-s − 9-s + (0.818 − 1.15i)10-s − 5.95·11-s + (1.88 − 0.658i)12-s − 3.87·13-s + (−3.59 − 1.05i)14-s + i·15-s + (−3.13 + 2.48i)16-s + 2.66i·17-s + ⋯
L(s)  = 1  + (0.579 − 0.815i)2-s + 0.577i·3-s + (−0.329 − 0.944i)4-s + 0.447·5-s + (0.470 + 0.334i)6-s + (−0.326 − 0.945i)7-s + (−0.960 − 0.278i)8-s − 0.333·9-s + (0.258 − 0.364i)10-s − 1.79·11-s + (0.545 − 0.190i)12-s − 1.07·13-s + (−0.959 − 0.281i)14-s + 0.258i·15-s + (−0.783 + 0.621i)16-s + 0.645i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0505i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0505i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.998 + 0.0505i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (811, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ -0.998 + 0.0505i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0249888 - 0.988081i\)
\(L(\frac12)\) \(\approx\) \(0.0249888 - 0.988081i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.818 + 1.15i)T \)
3 \( 1 - iT \)
5 \( 1 - T \)
7 \( 1 + (0.864 + 2.50i)T \)
good11 \( 1 + 5.95T + 11T^{2} \)
13 \( 1 + 3.87T + 13T^{2} \)
17 \( 1 - 2.66iT - 17T^{2} \)
19 \( 1 + 6.21iT - 19T^{2} \)
23 \( 1 + 4.60iT - 23T^{2} \)
29 \( 1 - 0.579iT - 29T^{2} \)
31 \( 1 - 5.65T + 31T^{2} \)
37 \( 1 + 2.21iT - 37T^{2} \)
41 \( 1 - 5.75iT - 41T^{2} \)
43 \( 1 - 5.16T + 43T^{2} \)
47 \( 1 - 5.39T + 47T^{2} \)
53 \( 1 + 10.3iT - 53T^{2} \)
59 \( 1 + 10.7iT - 59T^{2} \)
61 \( 1 + 10.9T + 61T^{2} \)
67 \( 1 + 10.0T + 67T^{2} \)
71 \( 1 + 0.496iT - 71T^{2} \)
73 \( 1 - 2.04iT - 73T^{2} \)
79 \( 1 + 10.6iT - 79T^{2} \)
83 \( 1 - 11.3iT - 83T^{2} \)
89 \( 1 - 10.8iT - 89T^{2} \)
97 \( 1 + 16.2iT - 97T^{2} \)
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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

−10.07263710606150090631661996249, −9.377739592813401274224461887477, −8.224458224618475285070716910327, −7.07741520056293009151466943630, −6.02328401790740826934224728240, −4.92298858679093232332036340716, −4.51627340710931781198339151302, −3.09882701288990139336143594953, −2.37725608078563351635355586658, −0.36225839333860030260676655071, 2.36151848523816621624255892089, 3.03086084713619132152502138328, 4.71810752666641852955293763051, 5.61775777324179982139030930323, 5.98384197483625242059179900829, 7.34551861878859882681827706413, 7.73177715800347237764063501031, 8.722888377124153038105097885286, 9.611131567938361612960480443454, 10.51169489472078774873218145731

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