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Properties

Label 2-1960-7.4-c1-0-7
Degree $2$
Conductor $1960$
Sign $0.198 - 0.980i$
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  + (1.09 − 1.89i)3-s + (0.5 + 0.866i)5-s + (−0.884 − 1.53i)9-s + (−1.79 + 3.11i)11-s − 5.85·13-s + 2.18·15-s + (−3.18 + 5.50i)17-s + (2.25 + 3.89i)19-s + (−1.31 − 2.27i)23-s + (−0.499 + 0.866i)25-s + 2.68·27-s + 2.05·29-s + (−3.31 + 5.74i)31-s + (3.92 + 6.80i)33-s + (2.95 + 5.12i)37-s + ⋯
L(s)  = 1  + (0.630 − 1.09i)3-s + (0.223 + 0.387i)5-s + (−0.294 − 0.510i)9-s + (−0.542 + 0.939i)11-s − 1.62·13-s + 0.563·15-s + (−0.771 + 1.33i)17-s + (0.516 + 0.894i)19-s + (−0.274 − 0.474i)23-s + (−0.0999 + 0.173i)25-s + 0.517·27-s + 0.382·29-s + (−0.595 + 1.03i)31-s + (0.683 + 1.18i)33-s + (0.486 + 0.842i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.198 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1960\)    =    \(2^{3} \cdot 5 \cdot 7^{2}\)
Sign: $0.198 - 0.980i$
Analytic conductor: \(15.6506\)
Root analytic conductor: \(3.95609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1960} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1960,\ (\ :1/2),\ 0.198 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.254027378\)
\(L(\frac12)\) \(\approx\) \(1.254027378\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 \)
good3 \( 1 + (-1.09 + 1.89i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (1.79 - 3.11i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 5.85T + 13T^{2} \)
17 \( 1 + (3.18 - 5.50i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.25 - 3.89i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.31 + 2.27i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.05T + 29T^{2} \)
31 \( 1 + (3.31 - 5.74i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.95 - 5.12i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 7.22T + 41T^{2} \)
43 \( 1 + 5.34T + 43T^{2} \)
47 \( 1 + (-1.15 - 2.00i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.05 + 3.55i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.53 - 4.39i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.68 + 6.38i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.69 - 2.94i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 16.7T + 71T^{2} \)
73 \( 1 + (-7.35 + 12.7i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.62 - 2.81i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 10.6T + 83T^{2} \)
89 \( 1 + (-5.57 - 9.65i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 17.1T + 97T^{2} \)
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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

−9.341934243461166084187906118016, −8.199475656080279973634665858483, −7.900714529620030526198962002588, −6.92804690424484454889421714810, −6.61787614389440414339973448392, −5.34623560835177816200805831031, −4.50334621126362185238593880688, −3.19919382746954116144772738026, −2.25120854891636335826180015051, −1.68303686551565594999127367243, 0.37826377385708166208727826888, 2.38456745802007765101027059369, 3.01531344685477473723423891232, 4.11178077219895965117829240390, 4.97615397683311583423667160520, 5.39287041539734013680371986405, 6.75322745818300691073231200065, 7.56112643382100999909348635120, 8.425246009909684634505450296824, 9.299156959695789399402322376463

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