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Properties

Label 2-1920-12.11-c1-0-19
Degree $2$
Conductor $1920$
Sign $0.356 - 0.934i$
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  + (−0.618 + 1.61i)3-s + i·5-s − 2i·7-s + (−2.23 − 2.00i)9-s + 2.47·11-s + 1.23·13-s + (−1.61 − 0.618i)15-s − 0.763i·17-s + 5.23i·19-s + (3.23 + 1.23i)21-s + 0.472·23-s − 25-s + (4.61 − 2.38i)27-s − 8.47i·29-s + 4.76i·31-s + ⋯
L(s)  = 1  + (−0.356 + 0.934i)3-s + 0.447i·5-s − 0.755i·7-s + (−0.745 − 0.666i)9-s + 0.745·11-s + 0.342·13-s + (−0.417 − 0.159i)15-s − 0.185i·17-s + 1.20i·19-s + (0.706 + 0.269i)21-s + 0.0984·23-s − 0.200·25-s + (0.888 − 0.458i)27-s − 1.57i·29-s + 0.855i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.356 - 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.356 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1920\)    =    \(2^{7} \cdot 3 \cdot 5\)
Sign: $0.356 - 0.934i$
Analytic conductor: \(15.3312\)
Root analytic conductor: \(3.91551\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1920} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1920,\ (\ :1/2),\ 0.356 - 0.934i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.524658499\)
\(L(\frac12)\) \(\approx\) \(1.524658499\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.618 - 1.61i)T \)
5 \( 1 - iT \)
good7 \( 1 + 2iT - 7T^{2} \)
11 \( 1 - 2.47T + 11T^{2} \)
13 \( 1 - 1.23T + 13T^{2} \)
17 \( 1 + 0.763iT - 17T^{2} \)
19 \( 1 - 5.23iT - 19T^{2} \)
23 \( 1 - 0.472T + 23T^{2} \)
29 \( 1 + 8.47iT - 29T^{2} \)
31 \( 1 - 4.76iT - 31T^{2} \)
37 \( 1 - 7.70T + 37T^{2} \)
41 \( 1 + 1.52iT - 41T^{2} \)
43 \( 1 - 9.70iT - 43T^{2} \)
47 \( 1 - 4.47T + 47T^{2} \)
53 \( 1 + 4.47iT - 53T^{2} \)
59 \( 1 - 6.47T + 59T^{2} \)
61 \( 1 - 12.4T + 61T^{2} \)
67 \( 1 - 11.2iT - 67T^{2} \)
71 \( 1 - 4T + 71T^{2} \)
73 \( 1 + 0.472T + 73T^{2} \)
79 \( 1 - 8.18iT - 79T^{2} \)
83 \( 1 + 11.7T + 83T^{2} \)
89 \( 1 - 1.52iT - 89T^{2} \)
97 \( 1 + 12.4T + 97T^{2} \)
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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

−9.687963597511723098973860301978, −8.623038536613229651034898456164, −7.85901279630310534247801194533, −6.82398750904941092109139357871, −6.16131530400902620773641473818, −5.35177240514990307255297511844, −4.08566541195703151691175913440, −3.90630619031656235645132869779, −2.66898222692323206581130604943, −1.01223335450960152705044685934, 0.75487618433219003998161834055, 1.89337015430623446756137943058, 2.87778221110115695290435046549, 4.18389345280658754818252238672, 5.24606143271271431790127939873, 5.85131608777129323898687340792, 6.72195971696560272052635887975, 7.34359476367446198920440958652, 8.432095716789926301089049614303, 8.839107277052600642613345239342

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