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Properties

Label 2-840-840.629-c1-0-159
Degree $2$
Conductor $840$
Sign $0.647 + 0.761i$
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  + (1.37 + 0.347i)2-s + (0.973 − 1.43i)3-s + (1.75 + 0.953i)4-s + (−1.31 − 1.80i)5-s + (1.83 − 1.62i)6-s + (2.61 + 0.406i)7-s + (2.07 + 1.91i)8-s + (−1.10 − 2.78i)9-s + (−1.17 − 2.93i)10-s − 2.00·11-s + (3.07 − 1.59i)12-s − 1.33i·13-s + (3.44 + 1.46i)14-s + (−3.87 + 0.131i)15-s + (2.17 + 3.35i)16-s − 4.12i·17-s + ⋯
L(s)  = 1  + (0.969 + 0.246i)2-s + (0.561 − 0.827i)3-s + (0.878 + 0.476i)4-s + (−0.589 − 0.807i)5-s + (0.748 − 0.663i)6-s + (0.988 + 0.153i)7-s + (0.734 + 0.678i)8-s + (−0.368 − 0.929i)9-s + (−0.372 − 0.927i)10-s − 0.605·11-s + (0.888 − 0.459i)12-s − 0.368i·13-s + (0.920 + 0.391i)14-s + (−0.999 + 0.0339i)15-s + (0.544 + 0.838i)16-s − 1.00i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.647 + 0.761i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.647 + 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.647 + 0.761i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (629, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ 0.647 + 0.761i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.01764 - 1.39491i\)
\(L(\frac12)\) \(\approx\) \(3.01764 - 1.39491i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.37 - 0.347i)T \)
3 \( 1 + (-0.973 + 1.43i)T \)
5 \( 1 + (1.31 + 1.80i)T \)
7 \( 1 + (-2.61 - 0.406i)T \)
good11 \( 1 + 2.00T + 11T^{2} \)
13 \( 1 + 1.33iT - 13T^{2} \)
17 \( 1 + 4.12iT - 17T^{2} \)
19 \( 1 - 2.33T + 19T^{2} \)
23 \( 1 - 0.0975T + 23T^{2} \)
29 \( 1 - 7.56T + 29T^{2} \)
31 \( 1 - 9.48iT - 31T^{2} \)
37 \( 1 + 6.19T + 37T^{2} \)
41 \( 1 + 5.88T + 41T^{2} \)
43 \( 1 + 3.65T + 43T^{2} \)
47 \( 1 + 7.79iT - 47T^{2} \)
53 \( 1 - 2.49iT - 53T^{2} \)
59 \( 1 - 1.86iT - 59T^{2} \)
61 \( 1 - 11.3T + 61T^{2} \)
67 \( 1 - 7.33T + 67T^{2} \)
71 \( 1 - 8.88iT - 71T^{2} \)
73 \( 1 + 6.85T + 73T^{2} \)
79 \( 1 - 10.9T + 79T^{2} \)
83 \( 1 + 10.1T + 83T^{2} \)
89 \( 1 + 13.4T + 89T^{2} \)
97 \( 1 + 9.65T + 97T^{2} \)
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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

−10.20590979703267961011006021092, −8.643514911547344968239227377636, −8.343000999838285634269741622724, −7.44334897449674849636730804904, −6.81972227122247255258352023950, −5.32304387884106630325207881485, −4.96679554663698407137631276160, −3.62731679886157596818686846700, −2.61090883194757292586882362232, −1.28503084849848930250803099088, 2.01160421287794872992583640913, 3.05224042239958449901576963305, 3.98404204239421667220085326863, 4.68349951168298567149105918459, 5.65086448792404571288082805914, 6.84516751891345031468320785209, 7.82445406468811916145931380859, 8.391341584011435328422551498375, 9.896711317088372075851911752886, 10.44188739580618694428596372642

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