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Properties

Label 2-1920-20.7-c1-0-26
Degree $2$
Conductor $1920$
Sign $0.408 - 0.912i$
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 + (1.25 + 1.84i)5-s + (0.660 − 0.660i)7-s + 1.00i·9-s − 1.06i·11-s + (1.61 − 1.61i)13-s + (−0.419 + 2.19i)15-s + (3.06 + 3.06i)17-s − 3.08·19-s + 0.933·21-s + (4.94 + 4.94i)23-s + (−1.84 + 4.64i)25-s + (−0.707 + 0.707i)27-s − 2.36i·29-s − 1.95i·31-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (0.561 + 0.827i)5-s + (0.249 − 0.249i)7-s + 0.333i·9-s − 0.321i·11-s + (0.448 − 0.448i)13-s + (−0.108 + 0.567i)15-s + (0.742 + 0.742i)17-s − 0.708·19-s + 0.203·21-s + (1.03 + 1.03i)23-s + (−0.368 + 0.929i)25-s + (−0.136 + 0.136i)27-s − 0.438i·29-s − 0.350i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.408 - 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.408 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1920\)    =    \(2^{7} \cdot 3 \cdot 5\)
Sign: $0.408 - 0.912i$
Analytic conductor: \(15.3312\)
Root analytic conductor: \(3.91551\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1920} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1920,\ (\ :1/2),\ 0.408 - 0.912i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.391023358\)
\(L(\frac12)\) \(\approx\) \(2.391023358\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.25 - 1.84i)T \)
good7 \( 1 + (-0.660 + 0.660i)T - 7iT^{2} \)
11 \( 1 + 1.06iT - 11T^{2} \)
13 \( 1 + (-1.61 + 1.61i)T - 13iT^{2} \)
17 \( 1 + (-3.06 - 3.06i)T + 17iT^{2} \)
19 \( 1 + 3.08T + 19T^{2} \)
23 \( 1 + (-4.94 - 4.94i)T + 23iT^{2} \)
29 \( 1 + 2.36iT - 29T^{2} \)
31 \( 1 + 1.95iT - 31T^{2} \)
37 \( 1 + (-4.38 - 4.38i)T + 37iT^{2} \)
41 \( 1 - 2.99T + 41T^{2} \)
43 \( 1 + (1.44 + 1.44i)T + 43iT^{2} \)
47 \( 1 + (-8.81 + 8.81i)T - 47iT^{2} \)
53 \( 1 + (5.15 - 5.15i)T - 53iT^{2} \)
59 \( 1 + 3.59T + 59T^{2} \)
61 \( 1 - 8.24T + 61T^{2} \)
67 \( 1 + (6.57 - 6.57i)T - 67iT^{2} \)
71 \( 1 + 7.24iT - 71T^{2} \)
73 \( 1 + (7.31 - 7.31i)T - 73iT^{2} \)
79 \( 1 - 3.45T + 79T^{2} \)
83 \( 1 + (1.74 + 1.74i)T + 83iT^{2} \)
89 \( 1 - 15.1iT - 89T^{2} \)
97 \( 1 + (11.0 + 11.0i)T + 97iT^{2} \)
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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

−9.446258000532030872447233252353, −8.565907767749737019037738259674, −7.82341546874930062146767267494, −7.06080948410993525266102672459, −6.03812401730040316174020860498, −5.49726835325824515636912466820, −4.25851076375055254942207082356, −3.41916068072593743267385802993, −2.61337912906969084278485728625, −1.36994418328955315113997618615, 0.922095859208662226436249558647, 1.96962592709993895776642907346, 2.91538025170933708711610444056, 4.23469149902158054332156749515, 4.97889766293811209853073372375, 5.88317578585704639838211080838, 6.69458605304643606217240963476, 7.56447396245709341780552692900, 8.437858359147473156960279877606, 8.992365013585613409659976214754

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