Properties

Label 2-1960-5.4-c1-0-17
Degree $2$
Conductor $1960$
Sign $0.447 - 0.894i$
Analytic cond. $15.6506$
Root an. cond. $3.95609$
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 − i)5-s + 2·9-s − 11-s i·13-s + (1 − 2i)15-s + 3i·17-s − 4·19-s − 2i·23-s + (3 + 4i)25-s + 5i·27-s + 29-s + 6·31-s i·33-s + 2i·37-s + ⋯
L(s)  = 1  + 0.577i·3-s + (−0.894 − 0.447i)5-s + 0.666·9-s − 0.301·11-s − 0.277i·13-s + (0.258 − 0.516i)15-s + 0.727i·17-s − 0.917·19-s − 0.417i·23-s + (0.600 + 0.800i)25-s + 0.962i·27-s + 0.185·29-s + 1.07·31-s − 0.174i·33-s + 0.328i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1960\)    =    \(2^{3} \cdot 5 \cdot 7^{2}\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(15.6506\)
Root analytic conductor: \(3.95609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1960} (1569, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1960,\ (\ :1/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.333883856\)
\(L(\frac12)\) \(\approx\) \(1.333883856\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (2 + i)T \)
7 \( 1 \)
good3 \( 1 - iT - 3T^{2} \)
11 \( 1 + T + 11T^{2} \)
13 \( 1 + iT - 13T^{2} \)
17 \( 1 - 3iT - 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + 2iT - 23T^{2} \)
29 \( 1 - T + 29T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 9iT - 47T^{2} \)
53 \( 1 - 14iT - 53T^{2} \)
59 \( 1 - 6T + 59T^{2} \)
61 \( 1 - 4T + 61T^{2} \)
67 \( 1 - 10iT - 67T^{2} \)
71 \( 1 + 16T + 71T^{2} \)
73 \( 1 - 10iT - 73T^{2} \)
79 \( 1 - 11T + 79T^{2} \)
83 \( 1 - 4iT - 83T^{2} \)
89 \( 1 - 12T + 89T^{2} \)
97 \( 1 - 19iT - 97T^{2} \)
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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

−9.222422968800396224412836216707, −8.536677345833429951867614839856, −7.87531002249553070606636250104, −7.06836063240099952703718723183, −6.12099001509499741255267768934, −5.06130092412691465277068948515, −4.30783774599762127389943430788, −3.79022341719444720541010835395, −2.54131173833178097868470506294, −1.02379750656731340639668614838, 0.60425216495940544797012983440, 2.05328371863018114736947674458, 3.06396650645317043376460911432, 4.15514975812277907812288964179, 4.78967635232577585948693001257, 6.13082086016116177721583784792, 6.78346923116214863583088022875, 7.52800915619795128245450998189, 8.004807196360362903996573534881, 8.942764804799642424343641100245

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