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Properties

Label 2-840-35.3-c1-0-2
Degree $2$
Conductor $840$
Sign $0.148 - 0.988i$
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 + (−2.21 − 0.285i)5-s + (0.311 − 2.62i)7-s + (−0.866 + 0.499i)9-s + (−2.32 + 4.02i)11-s + (−2.59 + 2.59i)13-s + (0.298 + 2.21i)15-s + (−1.24 + 0.333i)17-s + (3.67 + 6.36i)19-s + (−2.61 + 0.378i)21-s + (0.610 − 2.27i)23-s + (4.83 + 1.26i)25-s + (0.707 + 0.707i)27-s − 6.66i·29-s + (6.71 + 3.87i)31-s + ⋯
L(s)  = 1  + (−0.149 − 0.557i)3-s + (−0.991 − 0.127i)5-s + (0.117 − 0.993i)7-s + (−0.288 + 0.166i)9-s + (−0.701 + 1.21i)11-s + (−0.718 + 0.718i)13-s + (0.0770 + 0.572i)15-s + (−0.301 + 0.0808i)17-s + (0.842 + 1.45i)19-s + (−0.571 + 0.0826i)21-s + (0.127 − 0.474i)23-s + (0.967 + 0.253i)25-s + (0.136 + 0.136i)27-s − 1.23i·29-s + (1.20 + 0.696i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.148 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.148 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.148 - 0.988i$
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: \((2,\ 840,\ (\ :1/2),\ 0.148 - 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.460393 + 0.396303i\)
\(L(\frac12)\) \(\approx\) \(0.460393 + 0.396303i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.21 + 0.285i)T \)
7 \( 1 + (-0.311 + 2.62i)T \)
good11 \( 1 + (2.32 - 4.02i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.59 - 2.59i)T - 13iT^{2} \)
17 \( 1 + (1.24 - 0.333i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-3.67 - 6.36i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.610 + 2.27i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + 6.66iT - 29T^{2} \)
31 \( 1 + (-6.71 - 3.87i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.82 - 1.29i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 8.39iT - 41T^{2} \)
43 \( 1 + (7.18 + 7.18i)T + 43iT^{2} \)
47 \( 1 + (3.21 - 12.0i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (2.49 - 0.667i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (7.62 - 13.2i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.16 - 1.82i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.35 - 12.5i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 5.54T + 71T^{2} \)
73 \( 1 + (2.10 + 7.84i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (4.53 - 2.62i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.84 - 1.84i)T - 83iT^{2} \)
89 \( 1 + (6.67 + 11.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (9.98 + 9.98i)T + 97iT^{2} \)
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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

−10.28801905102638311512121881112, −9.791795801128431969115581480737, −8.394245084193225510294987697922, −7.62418901065433446222057877331, −7.26701883401940538166989431099, −6.24165364788247976758160949553, −4.73988163541293529952822677724, −4.30689654797971962176537429639, −2.87569821067381417344871224163, −1.38767637284350730306368468236, 0.31718981053224022863381517638, 2.75960625174054794818946531111, 3.34164466247138766110145565798, 4.86848152148612043809046215519, 5.29880882109087551986373522054, 6.49787404982910956179418060856, 7.62891125447797660328058431074, 8.363824927481322384246418072063, 9.083350530036500202008985287908, 10.02814844616555968652742592966

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