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Properties

Label 2-840-24.11-c1-0-7
Degree $2$
Conductor $840$
Sign $-0.871 - 0.491i$
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 + 1.39i)2-s + (−1.73 + 0.0660i)3-s + (−1.86 − 0.718i)4-s − 5-s + (0.355 − 2.42i)6-s − i·7-s + (1.48 − 2.40i)8-s + (2.99 − 0.228i)9-s + (0.258 − 1.39i)10-s + 1.12i·11-s + (3.27 + 1.12i)12-s − 3.42i·13-s + (1.39 + 0.258i)14-s + (1.73 − 0.0660i)15-s + (2.96 + 2.68i)16-s + 4.43i·17-s + ⋯
L(s)  = 1  + (−0.182 + 0.983i)2-s + (−0.999 + 0.0381i)3-s + (−0.933 − 0.359i)4-s − 0.447·5-s + (0.145 − 0.989i)6-s − 0.377i·7-s + (0.524 − 0.851i)8-s + (0.997 − 0.0762i)9-s + (0.0817 − 0.439i)10-s + 0.338i·11-s + (0.946 + 0.323i)12-s − 0.949i·13-s + (0.371 + 0.0691i)14-s + (0.446 − 0.0170i)15-s + (0.741 + 0.670i)16-s + 1.07i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.871 - 0.491i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.871 - 0.491i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.871 - 0.491i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (491, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ -0.871 - 0.491i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.132259 + 0.503833i\)
\(L(\frac12)\) \(\approx\) \(0.132259 + 0.503833i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 - 1.39i)T \)
3 \( 1 + (1.73 - 0.0660i)T \)
5 \( 1 + T \)
7 \( 1 + iT \)
good11 \( 1 - 1.12iT - 11T^{2} \)
13 \( 1 + 3.42iT - 13T^{2} \)
17 \( 1 - 4.43iT - 17T^{2} \)
19 \( 1 + 3.05T + 19T^{2} \)
23 \( 1 - 4.29T + 23T^{2} \)
29 \( 1 - 4.47T + 29T^{2} \)
31 \( 1 - 5.37iT - 31T^{2} \)
37 \( 1 - 2.22iT - 37T^{2} \)
41 \( 1 - 2.94iT - 41T^{2} \)
43 \( 1 + 12.9T + 43T^{2} \)
47 \( 1 + 9.04T + 47T^{2} \)
53 \( 1 - 2.89T + 53T^{2} \)
59 \( 1 - 5.77iT - 59T^{2} \)
61 \( 1 - 3.04iT - 61T^{2} \)
67 \( 1 + 6.06T + 67T^{2} \)
71 \( 1 - 12.5T + 71T^{2} \)
73 \( 1 - 0.929T + 73T^{2} \)
79 \( 1 - 15.8iT - 79T^{2} \)
83 \( 1 - 11.2iT - 83T^{2} \)
89 \( 1 - 11.5iT - 89T^{2} \)
97 \( 1 + 0.308T + 97T^{2} \)
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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

−10.40412373915545826631053406851, −9.895604113513481881603560170337, −8.575425083507264441881635218947, −7.948015666326345873236420388063, −6.89748595058356471192412985197, −6.42465044753260599419039714505, −5.29097178173204412637286908348, −4.63415853737010180736821052404, −3.58462347414298591070525157190, −1.16322295050446742643428654865, 0.37240334447885231586269678805, 1.89937903556129272248050903758, 3.29428693435100079660574844719, 4.48926680159028915035604873889, 5.07838491003423876169878070059, 6.33651359269282093085653038777, 7.26737615730305449916853545182, 8.363136039966954969147517630205, 9.215386341199243664091259769719, 9.997237511953508849757528911512

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