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Properties

Label 2-840-7.2-c1-0-9
Degree $2$
Conductor $840$
Sign $0.933 + 0.359i$
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  + (−0.5 − 0.866i)3-s + (0.5 − 0.866i)5-s + (2.63 − 0.245i)7-s + (−0.499 + 0.866i)9-s + (1.92 + 3.33i)11-s + 0.209·13-s − 0.999·15-s + (3.66 + 6.34i)17-s + (−0.5 + 0.866i)19-s + (−1.52 − 2.15i)21-s + (3.13 − 5.42i)23-s + (−0.499 − 0.866i)25-s + 0.999·27-s + 3.20·29-s + (−5.23 − 9.07i)31-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (0.223 − 0.387i)5-s + (0.995 − 0.0927i)7-s + (−0.166 + 0.288i)9-s + (0.580 + 1.00i)11-s + 0.0580·13-s − 0.258·15-s + (0.888 + 1.53i)17-s + (−0.114 + 0.198i)19-s + (−0.333 − 0.471i)21-s + (0.653 − 1.13i)23-s + (−0.0999 − 0.173i)25-s + 0.192·27-s + 0.595·29-s + (−0.940 − 1.62i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 + 0.359i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.933 + 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.933 + 0.359i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ 0.933 + 0.359i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69660 - 0.315178i\)
\(L(\frac12)\) \(\approx\) \(1.69660 - 0.315178i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-2.63 + 0.245i)T \)
good11 \( 1 + (-1.92 - 3.33i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 0.209T + 13T^{2} \)
17 \( 1 + (-3.66 - 6.34i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.13 + 5.42i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 3.20T + 29T^{2} \)
31 \( 1 + (5.23 + 9.07i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.95 - 6.84i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.26T + 41T^{2} \)
43 \( 1 - 2.84T + 43T^{2} \)
47 \( 1 + (-2.73 + 4.74i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.52 - 7.84i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.39 + 7.61i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.42 + 2.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 13.4T + 71T^{2} \)
73 \( 1 + (5.63 + 9.75i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.179 - 0.311i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.20T + 83T^{2} \)
89 \( 1 + (3.45 - 5.98i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 8T + 97T^{2} \)
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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

−10.29725744688120695999299184052, −9.245381613417619302231103717303, −8.315062903777343760135087761221, −7.69416457505088906283227115774, −6.67366999938954317010122977905, −5.79026545892419084644482179900, −4.81704341454822452068348368224, −3.95220296303094546051876300136, −2.16180235773029925661959994095, −1.24280034608058353820256524173, 1.17372716660844993512193863955, 2.84837170066588427473842552769, 3.83240029082583035434598238795, 5.14703525009035619164560642629, 5.57587757267864693719473669944, 6.84790989684157824880417747698, 7.62545181296939011658119925729, 8.796515333406797239709212011800, 9.304421093214761583102761833567, 10.39729107952065541595127879741

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