Properties

Label 2-840-840.797-c0-0-1
Degree $2$
Conductor $840$
Sign $0.229 - 0.973i$
Analytic cond. $0.419214$
Root an. cond. $0.647467$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 0.707i)2-s + (−0.382 − 0.923i)3-s − 1.00i·4-s + (0.382 + 0.923i)5-s + (0.923 + 0.382i)6-s + (0.707 + 0.707i)7-s + (0.707 + 0.707i)8-s + (−0.707 + 0.707i)9-s + (−0.923 − 0.382i)10-s + (−0.923 + 0.382i)12-s + (−1.30 + 1.30i)13-s − 1.00·14-s + (0.707 − 0.707i)15-s − 1.00·16-s i·18-s − 0.765i·19-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (−0.382 − 0.923i)3-s − 1.00i·4-s + (0.382 + 0.923i)5-s + (0.923 + 0.382i)6-s + (0.707 + 0.707i)7-s + (0.707 + 0.707i)8-s + (−0.707 + 0.707i)9-s + (−0.923 − 0.382i)10-s + (−0.923 + 0.382i)12-s + (−1.30 + 1.30i)13-s − 1.00·14-s + (0.707 − 0.707i)15-s − 1.00·16-s i·18-s − 0.765i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.229 - 0.973i$
Analytic conductor: \(0.419214\)
Root analytic conductor: \(0.647467\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (797, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :0),\ 0.229 - 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6243867529\)
\(L(\frac12)\) \(\approx\) \(0.6243867529\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (0.382 + 0.923i)T \)
5 \( 1 + (-0.382 - 0.923i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
good11 \( 1 + T^{2} \)
13 \( 1 + (1.30 - 1.30i)T - iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + 0.765iT - T^{2} \)
23 \( 1 + (-1 - i)T + iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 - 0.765T + T^{2} \)
61 \( 1 - 1.84T + T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + 1.41iT - T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + 1.41iT - T^{2} \)
83 \( 1 + (0.541 + 0.541i)T + iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - iT^{2} \)
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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

−10.59653593808114949687541551226, −9.520779880932448952371768638761, −8.893554792696618264178820115601, −7.76829920204580428348066715432, −7.12224533139881012306149312622, −6.57440665243773297820885064375, −5.54797725853779909750254374669, −4.84550912242740189153913146532, −2.55766681738740449173356675052, −1.75471889459808690040085823476, 0.872934332713651999777288495447, 2.55395093085680873567685738690, 3.88125212930056334834536621952, 4.77378272014413697342364915032, 5.45423428445852757565733926697, 6.99790430901327671843982117814, 8.095978429990034181111507114499, 8.618633601865618886710011815077, 9.733785380786163283069047174161, 10.10948903830152119357761729468

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