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Properties

Label 2-840-56.37-c1-0-38
Degree $2$
Conductor $840$
Sign $0.925 + 0.378i$
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  + (−1.36 + 0.366i)2-s + (0.866 − 0.5i)3-s + (1.73 − i)4-s + (0.866 + 0.5i)5-s + (−0.999 + i)6-s + (2 + 1.73i)7-s + (−1.99 + 2i)8-s + (0.499 − 0.866i)9-s + (−1.36 − 0.366i)10-s + (2.59 − 1.5i)11-s + (0.999 − 1.73i)12-s − 7i·13-s + (−3.36 − 1.63i)14-s + 0.999·15-s + (1.99 − 3.46i)16-s + (−3 − 5.19i)17-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)2-s + (0.499 − 0.288i)3-s + (0.866 − 0.5i)4-s + (0.387 + 0.223i)5-s + (−0.408 + 0.408i)6-s + (0.755 + 0.654i)7-s + (−0.707 + 0.707i)8-s + (0.166 − 0.288i)9-s + (−0.431 − 0.115i)10-s + (0.783 − 0.452i)11-s + (0.288 − 0.499i)12-s − 1.94i·13-s + (−0.899 − 0.436i)14-s + 0.258·15-s + (0.499 − 0.866i)16-s + (−0.727 − 1.26i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 + 0.378i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.925 + 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.925 + 0.378i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ 0.925 + 0.378i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41951 - 0.279298i\)
\(L(\frac12)\) \(\approx\) \(1.41951 - 0.279298i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 - 0.366i)T \)
3 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (-2 - 1.73i)T \)
good11 \( 1 + (-2.59 + 1.5i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 7iT - 13T^{2} \)
17 \( 1 + (3 + 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.59 - 1.5i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.5 - 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 6iT - 29T^{2} \)
31 \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-9.52 - 5.5i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 3T + 41T^{2} \)
43 \( 1 + 10iT - 43T^{2} \)
47 \( 1 + (-1.5 + 2.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (9.52 - 5.5i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.92 + 4i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.92 - 4i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.73 - i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 10T + 71T^{2} \)
73 \( 1 + (2 + 3.46i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + (7 - 12.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 4T + 97T^{2} \)
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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

−9.890116180245446210773314506203, −9.230844901990206016911832454618, −8.429967911492275989851509090373, −7.78782336720873392467207613805, −6.94877222126189560365451176563, −5.86631136236479688466028397956, −5.19978236503707098686423365680, −3.25680537872176439512421202100, −2.34099621577437718287573556452, −1.03981874665013207737782387729, 1.47016514491748912113033012802, 2.22343739721503689479485596504, 3.94305165912179434726597336809, 4.49232083042435405663238181585, 6.27931309959282877047039783775, 6.95459605796370952142378178825, 7.950074051288570606292710171529, 8.711496753424757659189750060367, 9.440852603645324939638262777137, 9.965232140111147422479019441900

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