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Properties

Label 2-1920-80.29-c1-0-17
Degree $2$
Conductor $1920$
Sign $0.397 - 0.917i$
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 + (2.15 + 0.607i)5-s − 2.25·7-s − 1.00i·9-s + (−1.66 + 1.66i)11-s + (4.76 − 4.76i)13-s + (−1.95 + 1.09i)15-s + 6.99i·17-s + (−2.66 − 2.66i)19-s + (1.59 − 1.59i)21-s + 4.41·23-s + (4.26 + 2.61i)25-s + (0.707 + 0.707i)27-s + (−2.59 − 2.59i)29-s + 3.93·31-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (0.962 + 0.271i)5-s − 0.851·7-s − 0.333i·9-s + (−0.500 + 0.500i)11-s + (1.32 − 1.32i)13-s + (−0.503 + 0.281i)15-s + 1.69i·17-s + (−0.611 − 0.611i)19-s + (0.347 − 0.347i)21-s + 0.921·23-s + (0.852 + 0.522i)25-s + (0.136 + 0.136i)27-s + (−0.481 − 0.481i)29-s + 0.706·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.397 - 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.397 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1920\)    =    \(2^{7} \cdot 3 \cdot 5\)
Sign: $0.397 - 0.917i$
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: \((2,\ 1920,\ (\ :1/2),\ 0.397 - 0.917i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.584357634\)
\(L(\frac12)\) \(\approx\) \(1.584357634\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.15 - 0.607i)T \)
good7 \( 1 + 2.25T + 7T^{2} \)
11 \( 1 + (1.66 - 1.66i)T - 11iT^{2} \)
13 \( 1 + (-4.76 + 4.76i)T - 13iT^{2} \)
17 \( 1 - 6.99iT - 17T^{2} \)
19 \( 1 + (2.66 + 2.66i)T + 19iT^{2} \)
23 \( 1 - 4.41T + 23T^{2} \)
29 \( 1 + (2.59 + 2.59i)T + 29iT^{2} \)
31 \( 1 - 3.93T + 31T^{2} \)
37 \( 1 + (-2.01 - 2.01i)T + 37iT^{2} \)
41 \( 1 - 4.50iT - 41T^{2} \)
43 \( 1 + (-7.14 - 7.14i)T + 43iT^{2} \)
47 \( 1 - 10.1iT - 47T^{2} \)
53 \( 1 + (0.649 + 0.649i)T + 53iT^{2} \)
59 \( 1 + (-5.64 + 5.64i)T - 59iT^{2} \)
61 \( 1 + (-5.00 - 5.00i)T + 61iT^{2} \)
67 \( 1 + (-4.95 + 4.95i)T - 67iT^{2} \)
71 \( 1 + 2.33iT - 71T^{2} \)
73 \( 1 + 2.18T + 73T^{2} \)
79 \( 1 + 6.38T + 79T^{2} \)
83 \( 1 + (5.25 - 5.25i)T - 83iT^{2} \)
89 \( 1 - 15.7iT - 89T^{2} \)
97 \( 1 + 4.61iT - 97T^{2} \)
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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

−9.578631888352380518078966053425, −8.640916130083566330975893703225, −7.88338468227457039489656580834, −6.60327660125250106115097366086, −6.16060409847544682103110725665, −5.54786234468575472452077069030, −4.47956672134132525705503518464, −3.41209856131712539551141339440, −2.59573295314462748016671139421, −1.13708036200417229196254462936, 0.70464631517790411300072539751, 1.95263229642581908252484486324, 2.97467388567299488480141477823, 4.13551018922784925794660556964, 5.28653224678900817696605620662, 5.86749455202400468780553130977, 6.69231449245865129332522463252, 7.15158100508361857879979621092, 8.551211585731473150728526152830, 9.003225984665264014403897028265

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