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Properties

Label 2-840-35.3-c1-0-20
Degree $2$
Conductor $840$
Sign $-0.117 + 0.993i$
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  + (−0.258 − 0.965i)3-s + (−0.394 − 2.20i)5-s + (2.44 + 1.01i)7-s + (−0.866 + 0.499i)9-s + (1.46 − 2.53i)11-s + (−0.741 + 0.741i)13-s + (−2.02 + 0.950i)15-s + (5.37 − 1.43i)17-s + (−0.502 − 0.870i)19-s + (0.343 − 2.62i)21-s + (1.12 − 4.18i)23-s + (−4.68 + 1.73i)25-s + (0.707 + 0.707i)27-s − 0.582i·29-s + (−1.56 − 0.902i)31-s + ⋯
L(s)  = 1  + (−0.149 − 0.557i)3-s + (−0.176 − 0.984i)5-s + (0.924 + 0.381i)7-s + (−0.288 + 0.166i)9-s + (0.442 − 0.765i)11-s + (−0.205 + 0.205i)13-s + (−0.522 + 0.245i)15-s + (1.30 − 0.349i)17-s + (−0.115 − 0.199i)19-s + (0.0749 − 0.572i)21-s + (0.233 − 0.872i)23-s + (−0.937 + 0.347i)25-s + (0.136 + 0.136i)27-s − 0.108i·29-s + (−0.280 − 0.162i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.117 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.117 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.117 + 0.993i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ -0.117 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00805 - 1.13451i\)
\(L(\frac12)\) \(\approx\) \(1.00805 - 1.13451i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.258 + 0.965i)T \)
5 \( 1 + (0.394 + 2.20i)T \)
7 \( 1 + (-2.44 - 1.01i)T \)
good11 \( 1 + (-1.46 + 2.53i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.741 - 0.741i)T - 13iT^{2} \)
17 \( 1 + (-5.37 + 1.43i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (0.502 + 0.870i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.12 + 4.18i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + 0.582iT - 29T^{2} \)
31 \( 1 + (1.56 + 0.902i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.12 + 0.836i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 4.95iT - 41T^{2} \)
43 \( 1 + (7.46 + 7.46i)T + 43iT^{2} \)
47 \( 1 + (-1.09 + 4.07i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (9.64 - 2.58i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-7.19 + 12.4i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.44 + 3.14i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.05 + 7.67i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 0.0421T + 71T^{2} \)
73 \( 1 + (-0.910 - 3.39i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (1.02 - 0.594i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.48 - 6.48i)T - 83iT^{2} \)
89 \( 1 + (-6.09 - 10.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.412 - 0.412i)T + 97iT^{2} \)
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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

−9.877588184392618916871327318953, −8.880379420433534793924827081987, −8.316659080214850575797639813962, −7.60169311408901742854425305934, −6.46088188721745469177376423763, −5.42247013361697653328481912565, −4.83663392168126215201206906825, −3.52337319245887556705865479819, −1.98838074577825889222540854592, −0.822046441886465858772238504222, 1.63680743469434762047145027708, 3.15835489598439618645507311571, 4.04225943220576644827115139871, 5.05082901656956057882978085487, 6.00981908928369024070136721493, 7.18679323112743535738898134704, 7.69704026766645417063982012079, 8.746739667796355218257145350827, 10.00838834839303393479535231922, 10.20966834800005969295505811073

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