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Properties

Label 2-840-840.53-c1-0-68
Degree $2$
Conductor $840$
Sign $-0.228 - 0.973i$
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  + (−1.40 + 0.120i)2-s + (1.62 + 0.585i)3-s + (1.97 − 0.339i)4-s + (1.18 + 1.89i)5-s + (−2.36 − 0.628i)6-s + (−0.390 + 2.61i)7-s + (−2.73 + 0.716i)8-s + (2.31 + 1.91i)9-s + (−1.90 − 2.52i)10-s + (−1.80 + 3.13i)11-s + (3.41 + 0.600i)12-s + (3.53 − 3.53i)13-s + (0.234 − 3.73i)14-s + (0.829 + 3.78i)15-s + (3.76 − 1.34i)16-s + (−0.454 + 0.121i)17-s + ⋯
L(s)  = 1  + (−0.996 + 0.0852i)2-s + (0.941 + 0.338i)3-s + (0.985 − 0.169i)4-s + (0.531 + 0.846i)5-s + (−0.966 − 0.256i)6-s + (−0.147 + 0.989i)7-s + (−0.967 + 0.253i)8-s + (0.771 + 0.636i)9-s + (−0.602 − 0.798i)10-s + (−0.545 + 0.945i)11-s + (0.984 + 0.173i)12-s + (0.981 − 0.981i)13-s + (0.0626 − 0.998i)14-s + (0.214 + 0.976i)15-s + (0.942 − 0.335i)16-s + (−0.110 + 0.0295i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.228 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.228 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.228 - 0.973i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ -0.228 - 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.909068 + 1.14707i\)
\(L(\frac12)\) \(\approx\) \(0.909068 + 1.14707i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 - 0.120i)T \)
3 \( 1 + (-1.62 - 0.585i)T \)
5 \( 1 + (-1.18 - 1.89i)T \)
7 \( 1 + (0.390 - 2.61i)T \)
good11 \( 1 + (1.80 - 3.13i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.53 + 3.53i)T - 13iT^{2} \)
17 \( 1 + (0.454 - 0.121i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.11 - 1.93i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.95 + 1.59i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 5.01iT - 29T^{2} \)
31 \( 1 + (3.14 - 5.45i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.03 + 7.60i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 3.45iT - 41T^{2} \)
43 \( 1 + (-8.67 + 8.67i)T - 43iT^{2} \)
47 \( 1 + (-1.02 + 3.81i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (1.80 - 0.482i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-9.17 - 5.29i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.15 - 3.55i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (13.6 - 3.64i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 1.40iT - 71T^{2} \)
73 \( 1 + (-7.71 + 2.06i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-13.2 + 7.66i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.69 - 2.69i)T + 83iT^{2} \)
89 \( 1 + (-3.58 - 6.20i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.12 + 4.12i)T - 97iT^{2} \)
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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

−10.38212693186504596317026590725, −9.536629904306178791596985729671, −8.870943472664128588065203887831, −7.972147907677643174281702456475, −7.37160139220080207286066962102, −6.19484784407086949249146181509, −5.46972524939168888883594350759, −3.66879149324272192414647192518, −2.60398258301703859981488018725, −1.96678610675622707879165312418, 0.894765121001375692198163184791, 1.89817026487186114241578866711, 3.23454742178016766658726651405, 4.26136949428767155711928149664, 5.93595704239143672460916718661, 6.69004876401682646538954927179, 7.78264053635966980955173439523, 8.272701893859642633612275238474, 9.195462799520681118360427780440, 9.593937548959820725293841073115

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