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Properties

Label 2-840-35.33-c1-0-10
Degree $2$
Conductor $840$
Sign $0.970 + 0.242i$
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  + (−0.965 − 0.258i)3-s + (1.42 − 1.72i)5-s + (2.63 − 0.206i)7-s + (0.866 + 0.499i)9-s + (2.47 + 4.29i)11-s + (0.0756 − 0.0756i)13-s + (−1.82 + 1.29i)15-s + (−0.738 + 2.75i)17-s + (−1.59 + 2.75i)19-s + (−2.60 − 0.483i)21-s + (7.11 − 1.90i)23-s + (−0.950 − 4.90i)25-s + (−0.707 − 0.707i)27-s − 9.21i·29-s + (−4.33 + 2.50i)31-s + ⋯
L(s)  = 1  + (−0.557 − 0.149i)3-s + (0.636 − 0.771i)5-s + (0.996 − 0.0779i)7-s + (0.288 + 0.166i)9-s + (0.746 + 1.29i)11-s + (0.0209 − 0.0209i)13-s + (−0.470 + 0.335i)15-s + (−0.179 + 0.668i)17-s + (−0.365 + 0.632i)19-s + (−0.567 − 0.105i)21-s + (1.48 − 0.397i)23-s + (−0.190 − 0.981i)25-s + (−0.136 − 0.136i)27-s − 1.71i·29-s + (−0.779 + 0.449i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.970 + 0.242i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (313, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ 0.970 + 0.242i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69129 - 0.208117i\)
\(L(\frac12)\) \(\approx\) \(1.69129 - 0.208117i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.965 + 0.258i)T \)
5 \( 1 + (-1.42 + 1.72i)T \)
7 \( 1 + (-2.63 + 0.206i)T \)
good11 \( 1 + (-2.47 - 4.29i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.0756 + 0.0756i)T - 13iT^{2} \)
17 \( 1 + (0.738 - 2.75i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.59 - 2.75i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-7.11 + 1.90i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 9.21iT - 29T^{2} \)
31 \( 1 + (4.33 - 2.50i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.118 - 0.440i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 7.00iT - 41T^{2} \)
43 \( 1 + (-5.76 - 5.76i)T + 43iT^{2} \)
47 \( 1 + (-6.57 + 1.76i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-2.19 + 8.19i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-1.25 - 2.17i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (8.22 + 4.74i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.63 + 0.706i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 11.8T + 71T^{2} \)
73 \( 1 + (-2.37 - 0.637i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (11.3 + 6.52i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.25 - 6.25i)T - 83iT^{2} \)
89 \( 1 + (-7.45 + 12.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.25 + 3.25i)T + 97iT^{2} \)
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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

−10.15252668585296119381513328261, −9.374568818991678293489760696398, −8.507658558771933405555910631981, −7.61623484474249444787067374784, −6.61705009032597534601498114731, −5.73884467859863723780489837284, −4.75597988400113900245474773910, −4.21097665255419938871494031648, −2.12650790793644005504387203830, −1.25310138928065460750242799863, 1.19152519864596521393248533049, 2.66024849310327137821063067734, 3.84106671398829494918253355826, 5.14825308439071607377520136910, 5.73177885698275873286158964765, 6.80619654226060354546448048727, 7.40987704486286465450460248539, 8.946431223865751646021879589707, 9.117245130778461257852678612839, 10.70016315923391459581314059792

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