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Properties

Label 2-840-280.67-c1-0-26
Degree $2$
Conductor $840$
Sign $0.552 - 0.833i$
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  + (−1.41 − 0.0812i)2-s + (−0.258 − 0.965i)3-s + (1.98 + 0.229i)4-s + (1.69 + 1.46i)5-s + (0.286 + 1.38i)6-s + (2.19 − 1.48i)7-s + (−2.78 − 0.485i)8-s + (−0.866 + 0.499i)9-s + (−2.26 − 2.20i)10-s + (−2.78 + 4.82i)11-s + (−0.292 − 1.97i)12-s + (4.22 + 4.22i)13-s + (−3.21 + 1.91i)14-s + (0.976 − 2.01i)15-s + (3.89 + 0.911i)16-s + (0.357 − 0.0957i)17-s + ⋯
L(s)  = 1  + (−0.998 − 0.0574i)2-s + (−0.149 − 0.557i)3-s + (0.993 + 0.114i)4-s + (0.755 + 0.654i)5-s + (0.117 + 0.565i)6-s + (0.827 − 0.560i)7-s + (−0.985 − 0.171i)8-s + (−0.288 + 0.166i)9-s + (−0.717 − 0.696i)10-s + (−0.839 + 1.45i)11-s + (−0.0845 − 0.571i)12-s + (1.17 + 1.17i)13-s + (−0.858 + 0.512i)14-s + (0.252 − 0.519i)15-s + (0.973 + 0.227i)16-s + (0.0866 − 0.0232i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.552 - 0.833i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.552 - 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.552 - 0.833i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ 0.552 - 0.833i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.906330 + 0.486360i\)
\(L(\frac12)\) \(\approx\) \(0.906330 + 0.486360i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 0.0812i)T \)
3 \( 1 + (0.258 + 0.965i)T \)
5 \( 1 + (-1.69 - 1.46i)T \)
7 \( 1 + (-2.19 + 1.48i)T \)
good11 \( 1 + (2.78 - 4.82i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.22 - 4.22i)T + 13iT^{2} \)
17 \( 1 + (-0.357 + 0.0957i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (3.88 - 2.24i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.37 - 5.12i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 2.15T + 29T^{2} \)
31 \( 1 + (7.57 + 4.37i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.48 + 0.398i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 6.48T + 41T^{2} \)
43 \( 1 + (4.80 - 4.80i)T - 43iT^{2} \)
47 \( 1 + (-1.25 - 0.335i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-1.73 + 0.463i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (2.91 + 1.68i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-9.74 + 5.62i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-13.9 + 3.73i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 10.8iT - 71T^{2} \)
73 \( 1 + (-1.90 - 7.09i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (0.655 + 1.13i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.80 + 6.80i)T - 83iT^{2} \)
89 \( 1 + (-3.23 + 1.86i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.31 - 5.31i)T + 97iT^{2} \)
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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

−10.25464777920168344101775224402, −9.648390777306442695071746264754, −8.574934119294438371888158423225, −7.72214658689649379865641188169, −7.03597463964970334812423659659, −6.36037272306638265629986981932, −5.27236124150743455116773008453, −3.76669823704583595578344942570, −2.06702109796601219534498174538, −1.68797066897270994830000294537, 0.70177870275054937608129869641, 2.18168067106757539336596466904, 3.36000095250507913571448155585, 5.11389520142842258031604232772, 5.66390510068653876002654703541, 6.44053605974821364707808114615, 8.053423465085794974446483916782, 8.635403187034652784320009957653, 8.842911441636714754320454762053, 10.27395609205690307400900839765

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