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Properties

Label 2-840-280.123-c1-0-90
Degree $2$
Conductor $840$
Sign $-0.709 + 0.704i$
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  + (−0.0812 + 1.41i)2-s + (0.965 − 0.258i)3-s + (−1.98 − 0.229i)4-s + (−0.422 − 2.19i)5-s + (0.286 + 1.38i)6-s + (−1.48 − 2.19i)7-s + (0.485 − 2.78i)8-s + (0.866 − 0.499i)9-s + (3.13 − 0.418i)10-s + (−2.78 + 4.82i)11-s + (−1.97 + 0.292i)12-s + (−4.22 + 4.22i)13-s + (3.21 − 1.91i)14-s + (−0.976 − 2.01i)15-s + (3.89 + 0.911i)16-s + (−0.0957 − 0.357i)17-s + ⋯
L(s)  = 1  + (−0.0574 + 0.998i)2-s + (0.557 − 0.149i)3-s + (−0.993 − 0.114i)4-s + (−0.188 − 0.981i)5-s + (0.117 + 0.565i)6-s + (−0.560 − 0.827i)7-s + (0.171 − 0.985i)8-s + (0.288 − 0.166i)9-s + (0.991 − 0.132i)10-s + (−0.839 + 1.45i)11-s + (−0.571 + 0.0845i)12-s + (−1.17 + 1.17i)13-s + (0.858 − 0.512i)14-s + (−0.252 − 0.519i)15-s + (0.973 + 0.227i)16-s + (−0.0232 − 0.0866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.709 + 0.704i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.709 + 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.709 + 0.704i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (403, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ -0.709 + 0.704i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0537483 - 0.130458i\)
\(L(\frac12)\) \(\approx\) \(0.0537483 - 0.130458i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0812 - 1.41i)T \)
3 \( 1 + (-0.965 + 0.258i)T \)
5 \( 1 + (0.422 + 2.19i)T \)
7 \( 1 + (1.48 + 2.19i)T \)
good11 \( 1 + (2.78 - 4.82i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.22 - 4.22i)T - 13iT^{2} \)
17 \( 1 + (0.0957 + 0.357i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-3.88 + 2.24i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.12 + 1.37i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 2.15T + 29T^{2} \)
31 \( 1 + (7.57 + 4.37i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.398 - 1.48i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 6.48T + 41T^{2} \)
43 \( 1 + (4.80 + 4.80i)T + 43iT^{2} \)
47 \( 1 + (-0.335 + 1.25i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-0.463 - 1.73i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-2.91 - 1.68i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-9.74 + 5.62i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.73 + 13.9i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 10.8iT - 71T^{2} \)
73 \( 1 + (7.09 - 1.90i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-0.655 - 1.13i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.80 - 6.80i)T + 83iT^{2} \)
89 \( 1 + (3.23 - 1.86i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.31 + 5.31i)T - 97iT^{2} \)
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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

−9.676993517424531237847172019608, −9.072725315333049773905859481761, −7.925516820529139760643726419725, −7.35401454392108715167015480709, −6.80374972144411716520985806963, −5.30611354222773438380619459970, −4.59154361261742283704617480181, −3.80503240677502682584919145918, −1.96283613938335362605282424182, −0.06147733437677313843153766294, 2.23133199665587206274216936843, 3.16106368794635377639892216085, 3.47260736364558524527555724626, 5.24129870642939661910901420255, 5.85915294721966292350298359302, 7.45524185299603417027323391942, 8.117649727151710968242741316754, 8.949936616483954807781364724438, 10.09675619826336921557193119586, 10.20322506467795690974346037206

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