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Properties

Label 2-840-40.29-c1-0-52
Degree $2$
Conductor $840$
Sign $-0.948 + 0.316i$
Analytic cond. $6.70743$
Root an. cond. $2.58987$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + i)2-s − 3-s − 2i·4-s + (−2 − i)5-s + (1 − i)6-s + i·7-s + (2 + 2i)8-s + 9-s + (3 − i)10-s − 4i·11-s + 2i·12-s + 2·13-s + (−1 − i)14-s + (2 + i)15-s − 4·16-s − 2i·17-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s − 0.577·3-s − i·4-s + (−0.894 − 0.447i)5-s + (0.408 − 0.408i)6-s + 0.377i·7-s + (0.707 + 0.707i)8-s + 0.333·9-s + (0.948 − 0.316i)10-s − 1.20i·11-s + 0.577i·12-s + 0.554·13-s + (−0.267 − 0.267i)14-s + (0.516 + 0.258i)15-s − 16-s − 0.485i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 + 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.948 + 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.948 + 0.316i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ -0.948 + 0.316i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - i)T \)
3 \( 1 + T \)
5 \( 1 + (2 + i)T \)
7 \( 1 - iT \)
good11 \( 1 + 4iT - 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 - 8iT - 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 + 2iT - 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + 6T + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 10T + 43T^{2} \)
47 \( 1 - 6iT - 47T^{2} \)
53 \( 1 + 14T + 53T^{2} \)
59 \( 1 - 6iT - 59T^{2} \)
61 \( 1 + 2iT - 61T^{2} \)
67 \( 1 - 10T + 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 + 14T + 79T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 - 8T + 89T^{2} \)
97 \( 1 + 6iT - 97T^{2} \)
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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

−9.734943247681086969613932545714, −8.651777371744057173275049665288, −8.275549062173679088523793607564, −7.36785276919967290597936632437, −6.26781392186522521072674815450, −5.66509429335177534367185896233, −4.67346177889569119987403938465, −3.45174096519215580512949610758, −1.40769865256966206039999480012, 0, 1.65603733781226436850893871696, 3.15260796033033971278292888267, 4.10000995042128240493500097903, 4.99064124873252736738842142605, 6.79315568196857936322539010217, 7.10135286658060362424950938518, 8.062441809345203619964575774025, 9.004404498362401074274100081351, 9.933622423645576485119565175719

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