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Properties

Label 2-840-21.17-c1-0-9
Degree $2$
Conductor $840$
Sign $-0.253 - 0.967i$
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  + (1.58 + 0.698i)3-s + (0.5 + 0.866i)5-s + (−2.26 + 1.37i)7-s + (2.02 + 2.21i)9-s + (−2.77 − 1.60i)11-s + 4.42i·13-s + (0.187 + 1.72i)15-s + (−2.76 + 4.78i)17-s + (−2.54 + 1.47i)19-s + (−4.54 + 0.593i)21-s + (6.13 − 3.54i)23-s + (−0.499 + 0.866i)25-s + (1.66 + 4.92i)27-s − 2.76i·29-s + (5.63 + 3.25i)31-s + ⋯
L(s)  = 1  + (0.915 + 0.403i)3-s + (0.223 + 0.387i)5-s + (−0.855 + 0.518i)7-s + (0.674 + 0.738i)9-s + (−0.836 − 0.482i)11-s + 1.22i·13-s + (0.0484 + 0.444i)15-s + (−0.670 + 1.16i)17-s + (−0.584 + 0.337i)19-s + (−0.991 + 0.129i)21-s + (1.27 − 0.738i)23-s + (−0.0999 + 0.173i)25-s + (0.319 + 0.947i)27-s − 0.513i·29-s + (1.01 + 0.584i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.253 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.253 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.253 - 0.967i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ -0.253 - 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04919 + 1.36015i\)
\(L(\frac12)\) \(\approx\) \(1.04919 + 1.36015i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.58 - 0.698i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (2.26 - 1.37i)T \)
good11 \( 1 + (2.77 + 1.60i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 4.42iT - 13T^{2} \)
17 \( 1 + (2.76 - 4.78i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.54 - 1.47i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-6.13 + 3.54i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.76iT - 29T^{2} \)
31 \( 1 + (-5.63 - 3.25i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.08 - 3.61i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 6.67T + 41T^{2} \)
43 \( 1 + 0.715T + 43T^{2} \)
47 \( 1 + (1.63 + 2.83i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-9.60 - 5.54i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.75 + 6.50i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.765 - 0.441i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.62 + 13.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.67iT - 71T^{2} \)
73 \( 1 + (2.16 + 1.24i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.95 - 8.57i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6.14T + 83T^{2} \)
89 \( 1 + (-3.12 - 5.41i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 0.313iT - 97T^{2} \)
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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

−10.35200778571965113742453270738, −9.573496595246930177294202882674, −8.706094848483586095596133105218, −8.246598324619806543859403192292, −6.86905369798216891119580784180, −6.31633009772292366241636420596, −4.99904019979188121561393733423, −3.93352457848033687766213217147, −2.92437896293571004904731666051, −2.07689722619240685782287412512, 0.73143876838293066044711675003, 2.46122341231353737228105463283, 3.20250551937190503692071389085, 4.47480630804513775407683122040, 5.51372668429715146264627564997, 6.82215890268196606247873421759, 7.33044508214467187998462882310, 8.290866725164793268066621052421, 9.109935827360261928241058859181, 9.868950364686848707270000355542

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