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Properties

Label 2-840-35.3-c1-0-5
Degree $2$
Conductor $840$
Sign $-0.862 - 0.505i$
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  + (0.258 + 0.965i)3-s + (−0.583 + 2.15i)5-s + (0.939 − 2.47i)7-s + (−0.866 + 0.499i)9-s + (−2.18 + 3.77i)11-s + (−2.79 + 2.79i)13-s + (−2.23 − 0.00489i)15-s + (−0.306 + 0.0821i)17-s + (−1.35 − 2.34i)19-s + (2.63 + 0.267i)21-s + (−0.783 + 2.92i)23-s + (−4.31 − 2.51i)25-s + (−0.707 − 0.707i)27-s + 4.52i·29-s + (1.89 + 1.09i)31-s + ⋯
L(s)  = 1  + (0.149 + 0.557i)3-s + (−0.260 + 0.965i)5-s + (0.355 − 0.934i)7-s + (−0.288 + 0.166i)9-s + (−0.657 + 1.13i)11-s + (−0.776 + 0.776i)13-s + (−0.577 − 0.00126i)15-s + (−0.0743 + 0.0199i)17-s + (−0.310 − 0.537i)19-s + (0.574 + 0.0583i)21-s + (−0.163 + 0.609i)23-s + (−0.863 − 0.503i)25-s + (−0.136 − 0.136i)27-s + 0.840i·29-s + (0.340 + 0.196i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.862 - 0.505i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.862 - 0.505i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.862 - 0.505i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ -0.862 - 0.505i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.253800 + 0.934881i\)
\(L(\frac12)\) \(\approx\) \(0.253800 + 0.934881i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.258 - 0.965i)T \)
5 \( 1 + (0.583 - 2.15i)T \)
7 \( 1 + (-0.939 + 2.47i)T \)
good11 \( 1 + (2.18 - 3.77i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.79 - 2.79i)T - 13iT^{2} \)
17 \( 1 + (0.306 - 0.0821i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.35 + 2.34i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.783 - 2.92i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 4.52iT - 29T^{2} \)
31 \( 1 + (-1.89 - 1.09i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.58 + 0.692i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 2.63iT - 41T^{2} \)
43 \( 1 + (-6.21 - 6.21i)T + 43iT^{2} \)
47 \( 1 + (-1.62 + 6.07i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (12.0 - 3.22i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (0.863 - 1.49i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (8.84 - 5.10i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.97 - 7.38i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 5.39T + 71T^{2} \)
73 \( 1 + (4.04 + 15.0i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (11.6 - 6.69i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-11.4 + 11.4i)T - 83iT^{2} \)
89 \( 1 + (-3.03 - 5.25i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.89 - 7.89i)T + 97iT^{2} \)
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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

−10.46013048742599275043771213108, −9.928685305243004794577004064953, −9.027099527759235567353537791327, −7.67843101202500828374009264397, −7.34119005869898599995316273353, −6.40173463712314699795825582811, −4.90371945008498983510555493681, −4.33961167167431871580561594096, −3.19236162021460233603514618968, −2.03918176553713858788988858596, 0.43949532802511641073674810809, 2.05700163396913184613141983493, 3.14111246105426207016709620571, 4.59413576632430157219586722134, 5.52439835939976966325398694359, 6.12380261166950954914403687258, 7.63789289427705987674208556841, 8.180103116410372917892058907476, 8.752578227603599880039590529618, 9.695468462375033532523036927325

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