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Properties

Label 2-840-840.653-c1-0-34
Degree $2$
Conductor $840$
Sign $-0.785 + 0.619i$
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  + (0.561 + 1.29i)2-s + (1.48 + 0.887i)3-s + (−1.37 + 1.45i)4-s + (−1.12 + 1.93i)5-s + (−0.317 + 2.42i)6-s + (−2.64 − 0.130i)7-s + (−2.66 − 0.960i)8-s + (1.42 + 2.64i)9-s + (−3.14 − 0.372i)10-s + (−1.45 − 2.51i)11-s + (−3.33 + 0.950i)12-s + (−1.13 + 1.13i)13-s + (−1.31 − 3.50i)14-s + (−3.38 + 1.87i)15-s + (−0.245 − 3.99i)16-s + (−1.55 + 5.80i)17-s + ⋯
L(s)  = 1  + (0.396 + 0.917i)2-s + (0.858 + 0.512i)3-s + (−0.685 + 0.728i)4-s + (−0.502 + 0.864i)5-s + (−0.129 + 0.991i)6-s + (−0.998 − 0.0493i)7-s + (−0.940 − 0.339i)8-s + (0.474 + 0.880i)9-s + (−0.993 − 0.117i)10-s + (−0.437 − 0.757i)11-s + (−0.961 + 0.274i)12-s + (−0.314 + 0.314i)13-s + (−0.351 − 0.936i)14-s + (−0.874 + 0.485i)15-s + (−0.0613 − 0.998i)16-s + (−0.377 + 1.40i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.785 + 0.619i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.785 + 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.785 + 0.619i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (653, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ -0.785 + 0.619i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.419987 - 1.21129i\)
\(L(\frac12)\) \(\approx\) \(0.419987 - 1.21129i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.561 - 1.29i)T \)
3 \( 1 + (-1.48 - 0.887i)T \)
5 \( 1 + (1.12 - 1.93i)T \)
7 \( 1 + (2.64 + 0.130i)T \)
good11 \( 1 + (1.45 + 2.51i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.13 - 1.13i)T - 13iT^{2} \)
17 \( 1 + (1.55 - 5.80i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-4.22 + 7.31i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.0569 - 0.212i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 5.64iT - 29T^{2} \)
31 \( 1 + (-3.40 - 5.89i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (6.70 - 1.79i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 4.91iT - 41T^{2} \)
43 \( 1 + (0.826 - 0.826i)T - 43iT^{2} \)
47 \( 1 + (4.89 - 1.31i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-1.85 + 6.91i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (0.222 - 0.128i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.0940 + 0.0542i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.83 - 14.3i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 12.8iT - 71T^{2} \)
73 \( 1 + (0.0731 - 0.272i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-0.639 - 0.369i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.53 - 2.53i)T + 83iT^{2} \)
89 \( 1 + (-0.894 + 1.54i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.84 - 7.84i)T - 97iT^{2} \)
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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

−10.51727376002339099746042183236, −9.731419484008391075364675159525, −8.766668045355287186400200475891, −8.228093112763208846419296797957, −7.06201421967284680788975074504, −6.73009506533007698309380673635, −5.43076594726569274836285088517, −4.34634779149219628995802156874, −3.31749679407344203306445184610, −2.89333616262423881990153042467, 0.47990716223334115754208194550, 1.99671554921712318525341972792, 3.08541709336019279254399459327, 3.93229845391451916873903967397, 4.96939442113468244804807998166, 6.03809567492422564674347423497, 7.33994752283130197615000739891, 8.038915361057255733146870073523, 9.174166718294135819728959884149, 9.603484211831360430761835173815

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