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Properties

Label 2-840-56.27-c1-0-28
Degree $2$
Conductor $840$
Sign $-0.366 - 0.930i$
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  + (1.25 + 0.655i)2-s + i·3-s + (1.13 + 1.64i)4-s + 5-s + (−0.655 + 1.25i)6-s + (0.658 + 2.56i)7-s + (0.349 + 2.80i)8-s − 9-s + (1.25 + 0.655i)10-s + 2.08·11-s + (−1.64 + 1.13i)12-s − 1.09·13-s + (−0.856 + 3.64i)14-s + i·15-s + (−1.40 + 3.74i)16-s − 5.02i·17-s + ⋯
L(s)  = 1  + (0.885 + 0.463i)2-s + 0.577i·3-s + (0.569 + 0.821i)4-s + 0.447·5-s + (−0.267 + 0.511i)6-s + (0.248 + 0.968i)7-s + (0.123 + 0.992i)8-s − 0.333·9-s + (0.396 + 0.207i)10-s + 0.630·11-s + (−0.474 + 0.328i)12-s − 0.304·13-s + (−0.228 + 0.973i)14-s + 0.258i·15-s + (−0.350 + 0.936i)16-s − 1.21i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.366 - 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.366 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.366 - 0.930i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (811, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ -0.366 - 0.930i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61448 + 2.37124i\)
\(L(\frac12)\) \(\approx\) \(1.61448 + 2.37124i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.25 - 0.655i)T \)
3 \( 1 - iT \)
5 \( 1 - T \)
7 \( 1 + (-0.658 - 2.56i)T \)
good11 \( 1 - 2.08T + 11T^{2} \)
13 \( 1 + 1.09T + 13T^{2} \)
17 \( 1 + 5.02iT - 17T^{2} \)
19 \( 1 + 0.996iT - 19T^{2} \)
23 \( 1 - 1.06iT - 23T^{2} \)
29 \( 1 - 1.45iT - 29T^{2} \)
31 \( 1 + 5.12T + 31T^{2} \)
37 \( 1 + 3.29iT - 37T^{2} \)
41 \( 1 + 2.56iT - 41T^{2} \)
43 \( 1 - 4.43T + 43T^{2} \)
47 \( 1 - 8.37T + 47T^{2} \)
53 \( 1 - 2.55iT - 53T^{2} \)
59 \( 1 - 4.32iT - 59T^{2} \)
61 \( 1 + 8.03T + 61T^{2} \)
67 \( 1 - 13.2T + 67T^{2} \)
71 \( 1 + 14.3iT - 71T^{2} \)
73 \( 1 - 2.39iT - 73T^{2} \)
79 \( 1 + 10.2iT - 79T^{2} \)
83 \( 1 - 10.8iT - 83T^{2} \)
89 \( 1 - 3.04iT - 89T^{2} \)
97 \( 1 + 18.9iT - 97T^{2} \)
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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

−10.63605055304740532337289220919, −9.274756474029133434314338419246, −9.011105576505888989731665036537, −7.75868849464278194119736266172, −6.86116718663752299270357337311, −5.81548551824273908414608788882, −5.24151379846862356818398292887, −4.33580055853945305964687208814, −3.14372760603526503589295521923, −2.16496270974955928033278687537, 1.15252056666024372898882803930, 2.15869737474768508175004386739, 3.55778224581825367065234907455, 4.38408027657303867331236439508, 5.53740209564223972096528444774, 6.39212800366145650798895057925, 7.09649485536666188389806334312, 8.079302434306973127523933212333, 9.300995283917791282215793212728, 10.21719903855177566492733130342

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