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Properties

Label 2-1920-80.27-c1-0-18
Degree $2$
Conductor $1920$
Sign $-0.301 - 0.953i$
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  + 3-s + (−0.311 + 2.21i)5-s + (1.96 + 1.96i)7-s + 9-s + (0.870 − 0.870i)11-s + 5.88i·13-s + (−0.311 + 2.21i)15-s + (2.69 + 2.69i)17-s + (−2.40 + 2.40i)19-s + (1.96 + 1.96i)21-s + (−2.63 + 2.63i)23-s + (−4.80 − 1.38i)25-s + 27-s + (−7.43 − 7.43i)29-s − 7.72i·31-s + ⋯
L(s)  = 1  + 0.577·3-s + (−0.139 + 0.990i)5-s + (0.743 + 0.743i)7-s + 0.333·9-s + (0.262 − 0.262i)11-s + 1.63i·13-s + (−0.0805 + 0.571i)15-s + (0.653 + 0.653i)17-s + (−0.550 + 0.550i)19-s + (0.429 + 0.429i)21-s + (−0.550 + 0.550i)23-s + (−0.961 − 0.276i)25-s + 0.192·27-s + (−1.38 − 1.38i)29-s − 1.38i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.301 - 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.301 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1920\)    =    \(2^{7} \cdot 3 \cdot 5\)
Sign: $-0.301 - 0.953i$
Analytic conductor: \(15.3312\)
Root analytic conductor: \(3.91551\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1920} (1567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1920,\ (\ :1/2),\ -0.301 - 0.953i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.082583996\)
\(L(\frac12)\) \(\approx\) \(2.082583996\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
5 \( 1 + (0.311 - 2.21i)T \)
good7 \( 1 + (-1.96 - 1.96i)T + 7iT^{2} \)
11 \( 1 + (-0.870 + 0.870i)T - 11iT^{2} \)
13 \( 1 - 5.88iT - 13T^{2} \)
17 \( 1 + (-2.69 - 2.69i)T + 17iT^{2} \)
19 \( 1 + (2.40 - 2.40i)T - 19iT^{2} \)
23 \( 1 + (2.63 - 2.63i)T - 23iT^{2} \)
29 \( 1 + (7.43 + 7.43i)T + 29iT^{2} \)
31 \( 1 + 7.72iT - 31T^{2} \)
37 \( 1 - 4.49iT - 37T^{2} \)
41 \( 1 + 4.84iT - 41T^{2} \)
43 \( 1 - 0.461iT - 43T^{2} \)
47 \( 1 + (-4.66 + 4.66i)T - 47iT^{2} \)
53 \( 1 - 2.41T + 53T^{2} \)
59 \( 1 + (-6.47 - 6.47i)T + 59iT^{2} \)
61 \( 1 + (8.50 - 8.50i)T - 61iT^{2} \)
67 \( 1 - 6.40iT - 67T^{2} \)
71 \( 1 - 13.3T + 71T^{2} \)
73 \( 1 + (1.62 + 1.62i)T + 73iT^{2} \)
79 \( 1 - 4.14T + 79T^{2} \)
83 \( 1 + 0.241T + 83T^{2} \)
89 \( 1 - 2.86T + 89T^{2} \)
97 \( 1 + (-3.18 - 3.18i)T + 97iT^{2} \)
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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

−9.420032613571115075297725589612, −8.593748266414481993129357610866, −7.899969414326850085837943906881, −7.22056773614331699060815225508, −6.22328879621593566472550349639, −5.64238559063406775515756560508, −4.14237384624075078798192280513, −3.77945467851532103676629720558, −2.34800771790981318951507319961, −1.84052127433078041385779762723, 0.70055403012347426823643630679, 1.73672441708686397074711775198, 3.10544295981811937155537504124, 4.00920495518528164198760317117, 4.92126303355386558689257256388, 5.45439514012007955592058482620, 6.80309985910141405161512634116, 7.74402341644673368911850434108, 8.031039859284266391062541763642, 8.942204768457092201738379497636

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