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Properties

Label 2-840-840.653-c1-0-149
Degree $2$
Conductor $840$
Sign $0.302 + 0.953i$
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  + (−0.120 + 1.40i)2-s + (0.585 + 1.62i)3-s + (−1.97 − 0.339i)4-s + (1.04 − 1.97i)5-s + (−2.36 + 0.628i)6-s + (−2.61 + 0.390i)7-s + (0.716 − 2.73i)8-s + (−2.31 + 1.91i)9-s + (2.65 + 1.71i)10-s + (−1.80 − 3.13i)11-s + (−0.600 − 3.41i)12-s + (−3.53 + 3.53i)13-s + (−0.234 − 3.73i)14-s + (3.83 + 0.545i)15-s + (3.76 + 1.34i)16-s + (−0.121 + 0.454i)17-s + ⋯
L(s)  = 1  + (−0.0852 + 0.996i)2-s + (0.338 + 0.941i)3-s + (−0.985 − 0.169i)4-s + (0.467 − 0.884i)5-s + (−0.966 + 0.256i)6-s + (−0.989 + 0.147i)7-s + (0.253 − 0.967i)8-s + (−0.771 + 0.636i)9-s + (0.840 + 0.541i)10-s + (−0.545 − 0.945i)11-s + (−0.173 − 0.984i)12-s + (−0.981 + 0.981i)13-s + (−0.0626 − 0.998i)14-s + (0.990 + 0.140i)15-s + (0.942 + 0.335i)16-s + (−0.0295 + 0.110i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.302 + 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.302 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.302 + 0.953i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (653, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ 0.302 + 0.953i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.158444 - 0.115910i\)
\(L(\frac12)\) \(\approx\) \(0.158444 - 0.115910i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.120 - 1.40i)T \)
3 \( 1 + (-0.585 - 1.62i)T \)
5 \( 1 + (-1.04 + 1.97i)T \)
7 \( 1 + (2.61 - 0.390i)T \)
good11 \( 1 + (1.80 + 3.13i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.53 - 3.53i)T - 13iT^{2} \)
17 \( 1 + (0.121 - 0.454i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.11 - 1.93i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.59 + 5.95i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 5.01iT - 29T^{2} \)
31 \( 1 + (3.14 + 5.45i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-7.60 + 2.03i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 3.45iT - 41T^{2} \)
43 \( 1 + (8.67 - 8.67i)T - 43iT^{2} \)
47 \( 1 + (-3.81 + 1.02i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-0.482 + 1.80i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (9.17 - 5.29i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.15 + 3.55i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.64 - 13.6i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 1.40iT - 71T^{2} \)
73 \( 1 + (2.06 - 7.71i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (13.2 + 7.66i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.69 - 2.69i)T + 83iT^{2} \)
89 \( 1 + (3.58 - 6.20i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.12 + 4.12i)T - 97iT^{2} \)
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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

−9.787826320457163804497563648975, −9.131506546090293207968378027254, −8.507390613280736209479495332210, −7.65542632747133017615104113552, −6.24634399608198560877884708227, −5.76194967079538425715179615794, −4.67817321782352720564326340682, −4.00518901473991787589051091570, −2.55038181657201746787093488222, −0.088383753256989359650451145166, 1.80134108619378815608079739995, 2.82381671081785197284683352012, 3.34735975002055543826883815147, 5.02488151149320558092312247255, 6.05591842262234724473037132002, 7.22970099689243239739564630682, 7.61222609057132379169143274324, 8.939204483344917957580455638606, 9.746623036460248065193273622880, 10.25440241815155071117650804197

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