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Properties

Label 2-840-21.17-c1-0-12
Degree $2$
Conductor $840$
Sign $0.998 - 0.0528i$
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  + (−0.789 − 1.54i)3-s + (0.5 + 0.866i)5-s + (−2.25 − 1.37i)7-s + (−1.75 + 2.43i)9-s + (3.85 + 2.22i)11-s + 5.71i·13-s + (0.940 − 1.45i)15-s + (0.510 − 0.884i)17-s + (4.42 − 2.55i)19-s + (−0.338 + 4.57i)21-s + (−0.712 + 0.411i)23-s + (−0.499 + 0.866i)25-s + (5.13 + 0.781i)27-s − 2.90i·29-s + (7.35 + 4.24i)31-s + ⋯
L(s)  = 1  + (−0.455 − 0.890i)3-s + (0.223 + 0.387i)5-s + (−0.854 − 0.520i)7-s + (−0.584 + 0.811i)9-s + (1.16 + 0.671i)11-s + 1.58i·13-s + (0.242 − 0.375i)15-s + (0.123 − 0.214i)17-s + (1.01 − 0.586i)19-s + (−0.0738 + 0.997i)21-s + (−0.148 + 0.0857i)23-s + (−0.0999 + 0.173i)25-s + (0.988 + 0.150i)27-s − 0.538i·29-s + (1.32 + 0.762i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0528i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0528i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.998 - 0.0528i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ 0.998 - 0.0528i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27554 + 0.0337271i\)
\(L(\frac12)\) \(\approx\) \(1.27554 + 0.0337271i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.789 + 1.54i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (2.25 + 1.37i)T \)
good11 \( 1 + (-3.85 - 2.22i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 5.71iT - 13T^{2} \)
17 \( 1 + (-0.510 + 0.884i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.42 + 2.55i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.712 - 0.411i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.90iT - 29T^{2} \)
31 \( 1 + (-7.35 - 4.24i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.77 - 3.07i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.21T + 41T^{2} \)
43 \( 1 + 0.722T + 43T^{2} \)
47 \( 1 + (5.90 + 10.2i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.04 - 0.601i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-7.29 + 12.6i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.34 + 0.777i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.55 - 13.0i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 14.3iT - 71T^{2} \)
73 \( 1 + (-13.7 - 7.92i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.31 - 2.27i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 0.0390T + 83T^{2} \)
89 \( 1 + (-1.60 - 2.77i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 1.20iT - 97T^{2} \)
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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

−10.02083484987655312101223397687, −9.566654144975358406822374188349, −8.502488302372406501714819747791, −7.16621196797516184345324722071, −6.84606834987143916325761939682, −6.22494985790423858581401316372, −4.90480599628908615729052119651, −3.79103248447815705951522979698, −2.44281673297824371990142728828, −1.17291942134494998345889318114, 0.818559222442617565866073192114, 2.98902608961134243177517413173, 3.69998518044208178863425384111, 4.94429510424241593455124482164, 5.95122092226281957561921785955, 6.20251238919133399392972619969, 7.78811043974029763368925122900, 8.768055640846751462419581741517, 9.468565978936507675788195060297, 10.06486349430152705709094412898

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