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Properties

Label 2-1920-80.43-c1-0-35
Degree $2$
Conductor $1920$
Sign $0.867 + 0.497i$
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  + i·3-s + (2.21 − 0.311i)5-s + (1.96 − 1.96i)7-s − 9-s + (0.870 − 0.870i)11-s − 5.88·13-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 − i·27-s + (7.43 + 7.43i)29-s − 7.72i·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + (0.990 − 0.139i)5-s + (0.743 − 0.743i)7-s − 0.333·9-s + (0.262 − 0.262i)11-s − 1.63·13-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 − 0.192i·27-s + (1.38 + 1.38i)29-s − 1.38i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.867 + 0.497i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.867 + 0.497i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1920\)    =    \(2^{7} \cdot 3 \cdot 5\)
Sign: $0.867 + 0.497i$
Analytic conductor: \(15.3312\)
Root analytic conductor: \(3.91551\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1920} (1183, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1920,\ (\ :1/2),\ 0.867 + 0.497i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.166247663\)
\(L(\frac12)\) \(\approx\) \(2.166247663\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
5 \( 1 + (-2.21 + 0.311i)T \)
good7 \( 1 + (-1.96 + 1.96i)T - 7iT^{2} \)
11 \( 1 + (-0.870 + 0.870i)T - 11iT^{2} \)
13 \( 1 + 5.88T + 13T^{2} \)
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 \( 1 - 4.49T + 37T^{2} \)
41 \( 1 + 4.84iT - 41T^{2} \)
43 \( 1 + 0.461T + 43T^{2} \)
47 \( 1 + (4.66 + 4.66i)T + 47iT^{2} \)
53 \( 1 - 2.41iT - 53T^{2} \)
59 \( 1 + (6.47 + 6.47i)T + 59iT^{2} \)
61 \( 1 + (8.50 - 8.50i)T - 61iT^{2} \)
67 \( 1 - 6.40T + 67T^{2} \)
71 \( 1 - 13.3T + 71T^{2} \)
73 \( 1 + (-1.62 + 1.62i)T - 73iT^{2} \)
79 \( 1 + 4.14T + 79T^{2} \)
83 \( 1 + 0.241iT - 83T^{2} \)
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

−9.395532620479034824949761587298, −8.428277741428632189549288878290, −7.54960220032266160413843960765, −6.83133809551522540929929851976, −5.76955277096818721214546221065, −4.91879312223845441015088789792, −4.52527436158071913136320902055, −3.14167252404163834023855351289, −2.21609844492129167540209866293, −0.834089179582470974242800405395, 1.39039427841848813898463094204, 2.19363546241880277139971694252, 3.04908145365840597930150686293, 4.62511570250312196711171177193, 5.32933656920027727883492378990, 6.07182919157574518248376912991, 6.83409490262361219246892511710, 7.85131990070899900883492054989, 8.267763685291795676944101959030, 9.500909077529116531193131092007

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