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Properties

Label 2-840-56.27-c1-0-33
Degree $2$
Conductor $840$
Sign $0.937 + 0.348i$
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  + (−1.35 + 0.402i)2-s − i·3-s + (1.67 − 1.09i)4-s + 5-s + (0.402 + 1.35i)6-s + (2.48 − 0.904i)7-s + (−1.83 + 2.15i)8-s − 9-s + (−1.35 + 0.402i)10-s + 5.85·11-s + (−1.09 − 1.67i)12-s + 0.623·13-s + (−3.00 + 2.22i)14-s − i·15-s + (1.61 − 3.65i)16-s + 1.19i·17-s + ⋯
L(s)  = 1  + (−0.958 + 0.284i)2-s − 0.577i·3-s + (0.837 − 0.545i)4-s + 0.447·5-s + (0.164 + 0.553i)6-s + (0.939 − 0.341i)7-s + (−0.647 + 0.761i)8-s − 0.333·9-s + (−0.428 + 0.127i)10-s + 1.76·11-s + (−0.315 − 0.483i)12-s + 0.172·13-s + (−0.803 + 0.595i)14-s − 0.258i·15-s + (0.404 − 0.914i)16-s + 0.288i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.937 + 0.348i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.937 + 0.348i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.937 + 0.348i$
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: \((2,\ 840,\ (\ :1/2),\ 0.937 + 0.348i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.33062 - 0.239195i\)
\(L(\frac12)\) \(\approx\) \(1.33062 - 0.239195i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.35 - 0.402i)T \)
3 \( 1 + iT \)
5 \( 1 - T \)
7 \( 1 + (-2.48 + 0.904i)T \)
good11 \( 1 - 5.85T + 11T^{2} \)
13 \( 1 - 0.623T + 13T^{2} \)
17 \( 1 - 1.19iT - 17T^{2} \)
19 \( 1 + 1.24iT - 19T^{2} \)
23 \( 1 - 6.12iT - 23T^{2} \)
29 \( 1 - 7.67iT - 29T^{2} \)
31 \( 1 - 1.68T + 31T^{2} \)
37 \( 1 + 3.34iT - 37T^{2} \)
41 \( 1 + 10.6iT - 41T^{2} \)
43 \( 1 + 8.65T + 43T^{2} \)
47 \( 1 + 4.59T + 47T^{2} \)
53 \( 1 - 4.55iT - 53T^{2} \)
59 \( 1 - 6.09iT - 59T^{2} \)
61 \( 1 - 7.66T + 61T^{2} \)
67 \( 1 - 4.21T + 67T^{2} \)
71 \( 1 + 1.86iT - 71T^{2} \)
73 \( 1 + 14.9iT - 73T^{2} \)
79 \( 1 - 5.19iT - 79T^{2} \)
83 \( 1 + 0.446iT - 83T^{2} \)
89 \( 1 + 17.2iT - 89T^{2} \)
97 \( 1 + 3.70iT - 97T^{2} \)
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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

−10.05343518898697144732369365431, −9.032212991409409059512275689589, −8.649573309676020556248675676218, −7.51328313361544454191776216326, −6.91942751158631691770150606455, −6.07622618776379643292604173036, −5.09610962680986530666460327069, −3.57807539431112925377674835666, −1.88346297506032229028841587609, −1.21098646340032486785105309573, 1.25336915726024955801515158146, 2.41871670734706302622095849797, 3.75937603109033100260682105244, 4.77735075240142836425224792513, 6.14922330539748871391379498291, 6.75202546377904278244876478495, 8.187666330698417206173043747948, 8.550353205459581624782675511216, 9.589353296464743879463990674659, 9.946349544339970405622678322794

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