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Properties

Label 2-1920-24.11-c1-0-48
Degree $2$
Conductor $1920$
Sign $0.945 + 0.325i$
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  + (0.759 + 1.55i)3-s + 5-s − 0.480i·7-s + (−1.84 + 2.36i)9-s − 4.73i·11-s − 1.59i·13-s + (0.759 + 1.55i)15-s − 6.32i·17-s − 2.09·19-s + (0.747 − 0.365i)21-s + 5.84·23-s + 25-s + (−5.08 − 1.07i)27-s + 8.95·29-s − 8.32i·31-s + ⋯
L(s)  = 1  + (0.438 + 0.898i)3-s + 0.447·5-s − 0.181i·7-s + (−0.615 + 0.788i)9-s − 1.42i·11-s − 0.441i·13-s + (0.196 + 0.401i)15-s − 1.53i·17-s − 0.481·19-s + (0.163 − 0.0796i)21-s + 1.21·23-s + 0.200·25-s + (−0.978 − 0.207i)27-s + 1.66·29-s − 1.49i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.325i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.945 + 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1920\)    =    \(2^{7} \cdot 3 \cdot 5\)
Sign: $0.945 + 0.325i$
Analytic conductor: \(15.3312\)
Root analytic conductor: \(3.91551\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1920} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1920,\ (\ :1/2),\ 0.945 + 0.325i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.060021542\)
\(L(\frac12)\) \(\approx\) \(2.060021542\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.759 - 1.55i)T \)
5 \( 1 - T \)
good7 \( 1 + 0.480iT - 7T^{2} \)
11 \( 1 + 4.73iT - 11T^{2} \)
13 \( 1 + 1.59iT - 13T^{2} \)
17 \( 1 + 6.32iT - 17T^{2} \)
19 \( 1 + 2.09T + 19T^{2} \)
23 \( 1 - 5.84T + 23T^{2} \)
29 \( 1 - 8.95T + 29T^{2} \)
31 \( 1 + 8.32iT - 31T^{2} \)
37 \( 1 - 3.67iT - 37T^{2} \)
41 \( 1 + 4.53iT - 41T^{2} \)
43 \( 1 + 8.70T + 43T^{2} \)
47 \( 1 - 0.578T + 47T^{2} \)
53 \( 1 - 7.26T + 53T^{2} \)
59 \( 1 + 2.30iT - 59T^{2} \)
61 \( 1 - 13.9iT - 61T^{2} \)
67 \( 1 - 0.706T + 67T^{2} \)
71 \( 1 + 13.6T + 71T^{2} \)
73 \( 1 - 1.80T + 73T^{2} \)
79 \( 1 - 6.54iT - 79T^{2} \)
83 \( 1 - 3.92iT - 83T^{2} \)
89 \( 1 + 9.26iT - 89T^{2} \)
97 \( 1 + 1.18T + 97T^{2} \)
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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

−8.985450458267795951215279960422, −8.693433312134668224488737301114, −7.76059857584686318769123316512, −6.77290510815330398886114838622, −5.76174143676295583973347581520, −5.11893822355432250321208683813, −4.22051640358629081459023967887, −3.11033727986350076406636283310, −2.59949082965905443143750987374, −0.74516938252246295847371292070, 1.38012991765055483398356695935, 2.11780484868781863952851613563, 3.13462557778124715104195542668, 4.32801036185626755490892469829, 5.24964427997305232869122272310, 6.43140697226439490203414730822, 6.74807423175244077487650918213, 7.63534106921996191926560643305, 8.588336326911698204357684071479, 8.968897231782212540715342645466

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