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Properties

Label 2-1920-80.29-c1-0-21
Degree $2$
Conductor $1920$
Sign $0.865 - 0.500i$
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 + (0.466 + 2.18i)5-s + 1.00·7-s − 1.00i·9-s + (−1.89 + 1.89i)11-s + (2.65 − 2.65i)13-s + (1.87 + 1.21i)15-s − 1.73i·17-s + (5.33 + 5.33i)19-s + (0.707 − 0.707i)21-s + 0.160·23-s + (−4.56 + 2.04i)25-s + (−0.707 − 0.707i)27-s + (−2.70 − 2.70i)29-s + 4.64·31-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (0.208 + 0.977i)5-s + 0.378·7-s − 0.333i·9-s + (−0.571 + 0.571i)11-s + (0.737 − 0.737i)13-s + (0.484 + 0.314i)15-s − 0.421i·17-s + (1.22 + 1.22i)19-s + (0.154 − 0.154i)21-s + 0.0334·23-s + (−0.912 + 0.408i)25-s + (−0.136 − 0.136i)27-s + (−0.501 − 0.501i)29-s + 0.833·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.865 - 0.500i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.865 - 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1920\)    =    \(2^{7} \cdot 3 \cdot 5\)
Sign: $0.865 - 0.500i$
Analytic conductor: \(15.3312\)
Root analytic conductor: \(3.91551\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1920} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1920,\ (\ :1/2),\ 0.865 - 0.500i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.238205912\)
\(L(\frac12)\) \(\approx\) \(2.238205912\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.466 - 2.18i)T \)
good7 \( 1 - 1.00T + 7T^{2} \)
11 \( 1 + (1.89 - 1.89i)T - 11iT^{2} \)
13 \( 1 + (-2.65 + 2.65i)T - 13iT^{2} \)
17 \( 1 + 1.73iT - 17T^{2} \)
19 \( 1 + (-5.33 - 5.33i)T + 19iT^{2} \)
23 \( 1 - 0.160T + 23T^{2} \)
29 \( 1 + (2.70 + 2.70i)T + 29iT^{2} \)
31 \( 1 - 4.64T + 31T^{2} \)
37 \( 1 + (-5.35 - 5.35i)T + 37iT^{2} \)
41 \( 1 + 9.89iT - 41T^{2} \)
43 \( 1 + (-7.23 - 7.23i)T + 43iT^{2} \)
47 \( 1 - 4.79iT - 47T^{2} \)
53 \( 1 + (-3.44 - 3.44i)T + 53iT^{2} \)
59 \( 1 + (-0.101 + 0.101i)T - 59iT^{2} \)
61 \( 1 + (-6.01 - 6.01i)T + 61iT^{2} \)
67 \( 1 + (9.04 - 9.04i)T - 67iT^{2} \)
71 \( 1 - 4.60iT - 71T^{2} \)
73 \( 1 - 12.1T + 73T^{2} \)
79 \( 1 - 5.73T + 79T^{2} \)
83 \( 1 + (2.04 - 2.04i)T - 83iT^{2} \)
89 \( 1 + 15.0iT - 89T^{2} \)
97 \( 1 + 3.84iT - 97T^{2} \)
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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

−9.403441085076736587396080427880, −8.176540388808855963282701642995, −7.75371505225997158592484974213, −7.08304048493452950514416231796, −6.06278285409631483775875691338, −5.47230773773524016218616311134, −4.18321535088980739268503479341, −3.17387873714454894302530085196, −2.46138088181939671570393914886, −1.24229350196146872724593264295, 0.900599936370790433191237974779, 2.12229924659678822942429832622, 3.29750497333051851148356039825, 4.27210365080530035274865696637, 5.04313480567395759520238533693, 5.72294480884701651065391037283, 6.77902440178253907675172915402, 7.905960353654650846282503727583, 8.359411279108632703405752781616, 9.257855519993251316435775395243

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