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Properties

Label 2-840-40.29-c1-0-47
Degree $2$
Conductor $840$
Sign $0.908 - 0.417i$
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 + 0.0297i)2-s + 3-s + (1.99 + 0.0840i)4-s + (1.06 + 1.96i)5-s + (1.41 + 0.0297i)6-s − i·7-s + (2.82 + 0.178i)8-s + 9-s + (1.44 + 2.81i)10-s + 1.23i·11-s + (1.99 + 0.0840i)12-s − 3.20·13-s + (0.0297 − 1.41i)14-s + (1.06 + 1.96i)15-s + (3.98 + 0.335i)16-s − 0.832i·17-s + ⋯
L(s)  = 1  + (0.999 + 0.0210i)2-s + 0.577·3-s + (0.999 + 0.0420i)4-s + (0.474 + 0.880i)5-s + (0.577 + 0.0121i)6-s − 0.377i·7-s + (0.998 + 0.0630i)8-s + 0.333·9-s + (0.455 + 0.890i)10-s + 0.373i·11-s + (0.576 + 0.0242i)12-s − 0.888·13-s + (0.00794 − 0.377i)14-s + (0.273 + 0.508i)15-s + (0.996 + 0.0839i)16-s − 0.201i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 - 0.417i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.908 - 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.908 - 0.417i$
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: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ 0.908 - 0.417i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.67631 + 0.805030i\)
\(L(\frac12)\) \(\approx\) \(3.67631 + 0.805030i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 - 0.0297i)T \)
3 \( 1 - T \)
5 \( 1 + (-1.06 - 1.96i)T \)
7 \( 1 + iT \)
good11 \( 1 - 1.23iT - 11T^{2} \)
13 \( 1 + 3.20T + 13T^{2} \)
17 \( 1 + 0.832iT - 17T^{2} \)
19 \( 1 - 2.35iT - 19T^{2} \)
23 \( 1 + 5.28iT - 23T^{2} \)
29 \( 1 + 4.68iT - 29T^{2} \)
31 \( 1 + 0.676T + 31T^{2} \)
37 \( 1 + 3.85T + 37T^{2} \)
41 \( 1 + 7.02T + 41T^{2} \)
43 \( 1 - 3.67T + 43T^{2} \)
47 \( 1 + 5.29iT - 47T^{2} \)
53 \( 1 + 1.25T + 53T^{2} \)
59 \( 1 - 2.45iT - 59T^{2} \)
61 \( 1 - 6.30iT - 61T^{2} \)
67 \( 1 - 15.6T + 67T^{2} \)
71 \( 1 + 5.86T + 71T^{2} \)
73 \( 1 + 6.52iT - 73T^{2} \)
79 \( 1 + 8.28T + 79T^{2} \)
83 \( 1 - 6.25T + 83T^{2} \)
89 \( 1 + 12.3T + 89T^{2} \)
97 \( 1 - 2.54iT - 97T^{2} \)
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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

−10.21291591757129227925163352591, −9.802391215217870231848850046813, −8.370996148940396919447157205675, −7.34335532597462801278473575882, −6.87841105488914175587844385869, −5.88270338021418131444200796438, −4.79498872766393804115412429283, −3.81553538593410745879945489171, −2.79000657120153013836438714986, −1.96209117516220122991377625560, 1.57477503949581089589373394112, 2.65240617318048335041751388376, 3.74525424876960681359150587374, 4.91169937294816738428589342214, 5.44407612709407694744109405317, 6.54907367092254535194257917783, 7.51554636510828425957827698215, 8.436303328284919351967732989815, 9.320021610405553656534083269847, 10.08215791671099453666716452737

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