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Properties

Label 2-840-840.53-c1-0-124
Degree $2$
Conductor $840$
Sign $0.753 - 0.657i$
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 + (1.67 − 0.448i)3-s + (1.73 + i)4-s + (−0.224 + 2.22i)5-s + 2.44·6-s + (0.358 − 2.62i)7-s + (1.99 + 2i)8-s + (2.59 − 1.50i)9-s + (−1.12 + 2.95i)10-s + (−3.18 + 5.50i)11-s + (3.34 + 0.896i)12-s + (1.44 − 3.44i)14-s + (0.621 + 3.82i)15-s + (1.99 + 3.46i)16-s + (4.09 − 1.09i)18-s + ⋯
L(s)  = 1  + (0.965 + 0.258i)2-s + (0.965 − 0.258i)3-s + (0.866 + 0.5i)4-s + (−0.100 + 0.994i)5-s + 0.999·6-s + (0.135 − 0.990i)7-s + (0.707 + 0.707i)8-s + (0.866 − 0.5i)9-s + (−0.354 + 0.935i)10-s + (−0.958 + 1.66i)11-s + (0.965 + 0.258i)12-s + (0.387 − 0.921i)14-s + (0.160 + 0.987i)15-s + (0.499 + 0.866i)16-s + (0.965 − 0.258i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.753 - 0.657i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.753 - 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.753 - 0.657i$
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: \((2,\ 840,\ (\ :1/2),\ 0.753 - 0.657i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.55640 + 1.33288i\)
\(L(\frac12)\) \(\approx\) \(3.55640 + 1.33288i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.67 + 0.448i)T \)
5 \( 1 + (0.224 - 2.22i)T \)
7 \( 1 + (-0.358 + 2.62i)T \)
good11 \( 1 + (3.18 - 5.50i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 13iT^{2} \)
17 \( 1 + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 10.7iT - 29T^{2} \)
31 \( 1 + (-5.55 + 9.62i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (13.5 - 3.61i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-7.13 - 4.11i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (16.2 - 4.35i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-1.13 + 0.655i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.60 - 8.60i)T + 83iT^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.95 + 4.95i)T - 97iT^{2} \)
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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

−10.15411441324921537756013882920, −9.814076058364624674766058095907, −7.968355602623651855115209081858, −7.67223329570748622693349412942, −6.99429193808006841194862979817, −6.13192681417448486670363123048, −4.55203497734984923899342959345, −4.03090927671500254410393873266, −2.80244813155749886653810754466, −2.07087685110150658189606124125, 1.48948633298325081871664476821, 2.83631172320366463302937882364, 3.46425629610109609378739147866, 4.89473615584296632925084231850, 5.26730351927156398965553989085, 6.39244298050948036409238307846, 7.75585472719762168612777544619, 8.532255273463021152002420573826, 9.043808231357068162712586728195, 10.22658493714295154488851561622

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