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Properties

Label 2-1920-80.69-c1-0-3
Degree $2$
Conductor $1920$
Sign $0.709 - 0.704i$
Analytic cond. $15.3312$
Root an. cond. $3.91551$
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  + (−0.707 − 0.707i)3-s + (−1.75 − 1.38i)5-s − 4.66·7-s + 1.00i·9-s + (−1.23 − 1.23i)11-s + (−4.12 − 4.12i)13-s + (0.258 + 2.22i)15-s − 3.20i·17-s + (−3.73 + 3.73i)19-s + (3.29 + 3.29i)21-s − 0.714·23-s + (1.14 + 4.86i)25-s + (0.707 − 0.707i)27-s + (1.24 − 1.24i)29-s + 3.84·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (−0.784 − 0.620i)5-s − 1.76·7-s + 0.333i·9-s + (−0.372 − 0.372i)11-s + (−1.14 − 1.14i)13-s + (0.0666 + 0.573i)15-s − 0.778i·17-s + (−0.857 + 0.857i)19-s + (0.719 + 0.719i)21-s − 0.149·23-s + (0.229 + 0.973i)25-s + (0.136 − 0.136i)27-s + (0.230 − 0.230i)29-s + 0.690·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.709 - 0.704i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.709 - 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1920\)    =    \(2^{7} \cdot 3 \cdot 5\)
Sign: $0.709 - 0.704i$
Analytic conductor: \(15.3312\)
Root analytic conductor: \(3.91551\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1920} (1249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1920,\ (\ :1/2),\ 0.709 - 0.704i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1887182805\)
\(L(\frac12)\) \(\approx\) \(0.1887182805\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (1.75 + 1.38i)T \)
good7 \( 1 + 4.66T + 7T^{2} \)
11 \( 1 + (1.23 + 1.23i)T + 11iT^{2} \)
13 \( 1 + (4.12 + 4.12i)T + 13iT^{2} \)
17 \( 1 + 3.20iT - 17T^{2} \)
19 \( 1 + (3.73 - 3.73i)T - 19iT^{2} \)
23 \( 1 + 0.714T + 23T^{2} \)
29 \( 1 + (-1.24 + 1.24i)T - 29iT^{2} \)
31 \( 1 - 3.84T + 31T^{2} \)
37 \( 1 + (-2.33 + 2.33i)T - 37iT^{2} \)
41 \( 1 + 6.81iT - 41T^{2} \)
43 \( 1 + (1.31 - 1.31i)T - 43iT^{2} \)
47 \( 1 + 1.18iT - 47T^{2} \)
53 \( 1 + (9.35 - 9.35i)T - 53iT^{2} \)
59 \( 1 + (-6.22 - 6.22i)T + 59iT^{2} \)
61 \( 1 + (-4.44 + 4.44i)T - 61iT^{2} \)
67 \( 1 + (6.37 + 6.37i)T + 67iT^{2} \)
71 \( 1 - 6.23iT - 71T^{2} \)
73 \( 1 - 5.34T + 73T^{2} \)
79 \( 1 + 13.0T + 79T^{2} \)
83 \( 1 + (-4.88 - 4.88i)T + 83iT^{2} \)
89 \( 1 - 2.20iT - 89T^{2} \)
97 \( 1 + 7.39iT - 97T^{2} \)
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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

−9.362122482364895920370601092882, −8.405440738635863245447802084484, −7.67127519481333579899766253962, −7.00887212694218811403369314409, −6.08099993888498732906186850106, −5.41174370613318454797232696452, −4.40018526295910903252230427736, −3.35077998834970305646536808314, −2.55854938343289584054759325185, −0.64061955293649963441413636244, 0.12139774298973584413515704622, 2.39297115842548336763261731056, 3.23742936162326852618794573922, 4.16653720568584302301729535090, 4.83866703597499745657271491305, 6.26655169675179013357885069256, 6.62206748293067068436984210830, 7.29147068044063813086673067778, 8.369750154948354200895030342675, 9.313365786630745927890756481271

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