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Properties

Label 2-840-35.13-c1-0-6
Degree $2$
Conductor $840$
Sign $0.633 - 0.773i$
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  + (−0.707 + 0.707i)3-s + (−0.600 + 2.15i)5-s + (−0.367 − 2.62i)7-s − 1.00i·9-s + 4.78·11-s + (0.368 − 0.368i)13-s + (−1.09 − 1.94i)15-s + (−1.10 − 1.10i)17-s + 5.13·19-s + (2.11 + 1.59i)21-s + (2.14 + 2.14i)23-s + (−4.27 − 2.58i)25-s + (0.707 + 0.707i)27-s + 9.26i·29-s + 8.98i·31-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (−0.268 + 0.963i)5-s + (−0.138 − 0.990i)7-s − 0.333i·9-s + 1.44·11-s + (0.102 − 0.102i)13-s + (−0.283 − 0.502i)15-s + (−0.268 − 0.268i)17-s + 1.17·19-s + (0.460 + 0.347i)21-s + (0.448 + 0.448i)23-s + (−0.855 − 0.517i)25-s + (0.136 + 0.136i)27-s + 1.72i·29-s + 1.61i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.633 - 0.773i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.633 - 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.633 - 0.773i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ 0.633 - 0.773i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22631 + 0.580716i\)
\(L(\frac12)\) \(\approx\) \(1.22631 + 0.580716i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (0.600 - 2.15i)T \)
7 \( 1 + (0.367 + 2.62i)T \)
good11 \( 1 - 4.78T + 11T^{2} \)
13 \( 1 + (-0.368 + 0.368i)T - 13iT^{2} \)
17 \( 1 + (1.10 + 1.10i)T + 17iT^{2} \)
19 \( 1 - 5.13T + 19T^{2} \)
23 \( 1 + (-2.14 - 2.14i)T + 23iT^{2} \)
29 \( 1 - 9.26iT - 29T^{2} \)
31 \( 1 - 8.98iT - 31T^{2} \)
37 \( 1 + (-3.69 + 3.69i)T - 37iT^{2} \)
41 \( 1 + 1.99iT - 41T^{2} \)
43 \( 1 + (1.62 + 1.62i)T + 43iT^{2} \)
47 \( 1 + (-7.81 - 7.81i)T + 47iT^{2} \)
53 \( 1 + (-0.588 - 0.588i)T + 53iT^{2} \)
59 \( 1 - 2.58T + 59T^{2} \)
61 \( 1 + 1.56iT - 61T^{2} \)
67 \( 1 + (6.06 - 6.06i)T - 67iT^{2} \)
71 \( 1 - 7.74T + 71T^{2} \)
73 \( 1 + (2.10 - 2.10i)T - 73iT^{2} \)
79 \( 1 + 4.57iT - 79T^{2} \)
83 \( 1 + (-6.55 + 6.55i)T - 83iT^{2} \)
89 \( 1 - 8.77T + 89T^{2} \)
97 \( 1 + (-0.711 - 0.711i)T + 97iT^{2} \)
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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

−10.48051683081908213930200415184, −9.581095481771049949336732382095, −8.837168213067738589648203540185, −7.35056532427540728545325638186, −7.02285713762759900184364179217, −6.09981971142973950749513470758, −4.88505353949197873343004624941, −3.80277557910490561498656481705, −3.19239003193224215253799156396, −1.18816000058293985895881593725, 0.891687608571707782156265141054, 2.21068090929998723833710437958, 3.79996754762648689403489805759, 4.77034652974091645010349964349, 5.80830051216102326408653480574, 6.41005473303773531754844938730, 7.60484659670927000786688250136, 8.446084592663607598475982405802, 9.240767739089976609061148745494, 9.792128727795357197536012646723

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