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Properties

Label 2-840-105.104-c1-0-19
Degree $2$
Conductor $840$
Sign $0.0872 - 0.996i$
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  + (0.726 + 1.57i)3-s + (2.20 − 0.341i)5-s + (2.06 + 1.64i)7-s + (−1.94 + 2.28i)9-s + 1.06i·11-s − 4.82·13-s + (2.14 + 3.22i)15-s + 7.89i·17-s − 4.02i·19-s + (−1.09 + 4.45i)21-s + 5.69·23-s + (4.76 − 1.50i)25-s + (−5.00 − 1.40i)27-s − 2.00i·29-s − 4.89i·31-s + ⋯
L(s)  = 1  + (0.419 + 0.907i)3-s + (0.988 − 0.152i)5-s + (0.781 + 0.623i)7-s + (−0.648 + 0.761i)9-s + 0.320i·11-s − 1.33·13-s + (0.552 + 0.833i)15-s + 1.91i·17-s − 0.922i·19-s + (−0.238 + 0.971i)21-s + 1.18·23-s + (0.953 − 0.301i)25-s + (−0.962 − 0.269i)27-s − 0.372i·29-s − 0.879i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0872 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0872 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.0872 - 0.996i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ 0.0872 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55951 + 1.42889i\)
\(L(\frac12)\) \(\approx\) \(1.55951 + 1.42889i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.726 - 1.57i)T \)
5 \( 1 + (-2.20 + 0.341i)T \)
7 \( 1 + (-2.06 - 1.64i)T \)
good11 \( 1 - 1.06iT - 11T^{2} \)
13 \( 1 + 4.82T + 13T^{2} \)
17 \( 1 - 7.89iT - 17T^{2} \)
19 \( 1 + 4.02iT - 19T^{2} \)
23 \( 1 - 5.69T + 23T^{2} \)
29 \( 1 + 2.00iT - 29T^{2} \)
31 \( 1 + 4.89iT - 31T^{2} \)
37 \( 1 - 2.56iT - 37T^{2} \)
41 \( 1 + 5.08T + 41T^{2} \)
43 \( 1 + 6.15iT - 43T^{2} \)
47 \( 1 - 2.27iT - 47T^{2} \)
53 \( 1 - 9.84T + 53T^{2} \)
59 \( 1 - 5.87T + 59T^{2} \)
61 \( 1 + 7.02iT - 61T^{2} \)
67 \( 1 - 10.8iT - 67T^{2} \)
71 \( 1 - 0.0512iT - 71T^{2} \)
73 \( 1 + 2.86T + 73T^{2} \)
79 \( 1 - 7.00T + 79T^{2} \)
83 \( 1 + 7.59iT - 83T^{2} \)
89 \( 1 + 9.72T + 89T^{2} \)
97 \( 1 - 1.77T + 97T^{2} \)
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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

−10.26793608378496318070874641244, −9.568182209860348350535644950276, −8.821400545130577668324304908144, −8.168968887006764760308322282324, −6.97165100281248269572756654711, −5.70731482194941115313687389757, −5.07687337531113406866858157598, −4.25041488023959569545858149397, −2.72484175417095189924855164209, −1.93311188295145983364589966268, 1.03850360931131012119290689616, 2.25093973525367085077327709791, 3.16695918369681955305949018363, 4.85645235372808016201949102744, 5.54511080461935979527907015950, 6.96749521339953236165606431242, 7.15966462173715911042937439903, 8.246954210231725403366057746598, 9.170432620655156686814907847711, 9.876144369530099420610984277472

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