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Properties

Label 2-840-280.109-c1-0-86
Degree $2$
Conductor $840$
Sign $-0.997 - 0.0750i$
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  + (−1.36 − 0.366i)2-s + (0.5 − 0.866i)3-s + (1.73 + i)4-s + (2.23 − 0.133i)5-s + (−1 + 0.999i)6-s + (−1.73 − 2i)7-s + (−1.99 − 2i)8-s + (−0.499 − 0.866i)9-s + (−3.09 − 0.633i)10-s + (−2.59 − 1.5i)11-s + (1.73 − 0.999i)12-s − 3·13-s + (1.63 + 3.36i)14-s + (1 − 1.99i)15-s + (1.99 + 3.46i)16-s + (−3.46 − 2i)17-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)2-s + (0.288 − 0.499i)3-s + (0.866 + 0.5i)4-s + (0.998 − 0.0599i)5-s + (−0.408 + 0.408i)6-s + (−0.654 − 0.755i)7-s + (−0.707 − 0.707i)8-s + (−0.166 − 0.288i)9-s + (−0.979 − 0.200i)10-s + (−0.783 − 0.452i)11-s + (0.499 − 0.288i)12-s − 0.832·13-s + (0.436 + 0.899i)14-s + (0.258 − 0.516i)15-s + (0.499 + 0.866i)16-s + (−0.840 − 0.485i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0750i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.997 - 0.0750i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ -0.997 - 0.0750i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0176198 + 0.469029i\)
\(L(\frac12)\) \(\approx\) \(0.0176198 + 0.469029i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 + 0.366i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (-2.23 + 0.133i)T \)
7 \( 1 + (1.73 + 2i)T \)
good11 \( 1 + (2.59 + 1.5i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 3T + 13T^{2} \)
17 \( 1 + (3.46 + 2i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (6.06 - 3.5i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (6.06 - 3.5i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 6iT - 29T^{2} \)
31 \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 7T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + (-11.2 + 6.5i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.5 + 7.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (10.3 + 6i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.73 + i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4 - 6.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10T + 71T^{2} \)
73 \( 1 + (-5.19 - 3i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (5 + 8.66i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 10T + 83T^{2} \)
89 \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 4iT - 97T^{2} \)
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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

−9.867288489272200236522634095663, −9.008982612158060167942210404452, −8.128536635372561372471842723733, −7.31780997989384140772343630341, −6.49571060868371265849180943319, −5.73472029360933224878962158526, −4.03577665797536006437863977154, −2.69712436497691054109150035013, −1.94198377503199700096132070601, −0.26361780288178110812720171150, 2.28201596817863885177423932094, 2.53823252430971857122902883068, 4.51123375840236032111205135098, 5.65128034893931087645521653785, 6.31714679676399014605013824632, 7.25722534602679486534162876469, 8.383567873835267701846531689726, 9.130871507460025332583092680156, 9.575965222514707412755374626652, 10.55559239411243392660656191243

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