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Properties

Label 2-840-21.17-c1-0-15
Degree $2$
Conductor $840$
Sign $0.994 - 0.107i$
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  + (1.51 − 0.838i)3-s + (0.5 + 0.866i)5-s + (−1.32 + 2.29i)7-s + (1.59 − 2.54i)9-s + (3.96 + 2.28i)11-s − 3.34i·13-s + (1.48 + 0.893i)15-s + (−1.18 + 2.06i)17-s + (1.42 − 0.821i)19-s + (−0.0858 + 4.58i)21-s + (5.45 − 3.14i)23-s + (−0.499 + 0.866i)25-s + (0.285 − 5.18i)27-s + 10.4i·29-s + (0.580 + 0.334i)31-s + ⋯
L(s)  = 1  + (0.875 − 0.484i)3-s + (0.223 + 0.387i)5-s + (−0.500 + 0.865i)7-s + (0.531 − 0.847i)9-s + (1.19 + 0.690i)11-s − 0.926i·13-s + (0.383 + 0.230i)15-s + (−0.288 + 0.499i)17-s + (0.326 − 0.188i)19-s + (−0.0187 + 0.999i)21-s + (1.13 − 0.656i)23-s + (−0.0999 + 0.173i)25-s + (0.0548 − 0.998i)27-s + 1.94i·29-s + (0.104 + 0.0601i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.107i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.994 - 0.107i$
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: \((2,\ 840,\ (\ :1/2),\ 0.994 - 0.107i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.23752 + 0.120973i\)
\(L(\frac12)\) \(\approx\) \(2.23752 + 0.120973i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.51 + 0.838i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (1.32 - 2.29i)T \)
good11 \( 1 + (-3.96 - 2.28i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.34iT - 13T^{2} \)
17 \( 1 + (1.18 - 2.06i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.42 + 0.821i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.45 + 3.14i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 10.4iT - 29T^{2} \)
31 \( 1 + (-0.580 - 0.334i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.44 - 4.23i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 7.20T + 41T^{2} \)
43 \( 1 - 0.255T + 43T^{2} \)
47 \( 1 + (-0.506 - 0.877i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.27 + 3.62i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.29 + 7.44i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (11.7 - 6.81i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.64 + 9.78i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.10iT - 71T^{2} \)
73 \( 1 + (11.9 + 6.89i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.43 + 11.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 5.90T + 83T^{2} \)
89 \( 1 + (-3.29 - 5.69i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 4.71iT - 97T^{2} \)
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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

−10.01008317313409157543535502808, −9.159992800515874649486114261381, −8.761016115473051437304431272985, −7.61877127388067817846158966635, −6.75930766946148430601590061397, −6.15503017295740063988351391932, −4.81676056723954708349112350635, −3.44416786111031428519441590053, −2.73064567011544650138660822924, −1.46937310323801830016633502209, 1.23113725006718568485207495841, 2.75262853914278529173410699714, 3.92596772254293747851199495785, 4.40945812125252308901980230724, 5.83074996109363473941023394328, 6.87456324650521450654184941212, 7.65414576439503853959807112589, 8.751897998617865166747756017200, 9.399197078937324769508195118137, 9.814974143274538807171100337523

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