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Properties

Label 2-1960-7.4-c1-0-5
Degree $2$
Conductor $1960$
Sign $-0.605 - 0.795i$
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  + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)5-s + (1 + 1.73i)9-s + (1 − 1.73i)11-s − 4·13-s + 0.999·15-s + (3 + 5.19i)19-s + (−1.5 − 2.59i)23-s + (−0.499 + 0.866i)25-s − 5·27-s − 3·29-s + (0.999 + 1.73i)33-s + (6 + 10.3i)37-s + (2 − 3.46i)39-s + 7·41-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (−0.223 − 0.387i)5-s + (0.333 + 0.577i)9-s + (0.301 − 0.522i)11-s − 1.10·13-s + 0.258·15-s + (0.688 + 1.19i)19-s + (−0.312 − 0.541i)23-s + (−0.0999 + 0.173i)25-s − 0.962·27-s − 0.557·29-s + (0.174 + 0.301i)33-s + (0.986 + 1.70i)37-s + (0.320 − 0.554i)39-s + 1.09·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1960\)    =    \(2^{3} \cdot 5 \cdot 7^{2}\)
Sign: $-0.605 - 0.795i$
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: \((2,\ 1960,\ (\ :1/2),\ -0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8936394117\)
\(L(\frac12)\) \(\approx\) \(0.8936394117\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3 - 5.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-6 - 10.3i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 7T + 41T^{2} \)
43 \( 1 + 9T + 43T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5 - 8.66i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.5 - 4.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.5 - 9.52i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 10T + 71T^{2} \)
73 \( 1 + (4 - 6.92i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3 + 5.19i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3T + 83T^{2} \)
89 \( 1 + (-8.5 - 14.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−9.704738507828606238765624889563, −8.642014564346220515520014263425, −7.896139070633674251882418481490, −7.25724552969849724084746020623, −6.12461488300044872172871879894, −5.35025823624114750546671489536, −4.59507761212462334650850989884, −3.85063913259806463374045667238, −2.67134640564900498057431619074, −1.35625031084951966351389565260, 0.34522657021726829050161345472, 1.77955254203218709510698288686, 2.87133031446484704671836476377, 3.94977614155180221790272179035, 4.83335997937364640788419283639, 5.82366969356590017387397967732, 6.68348294478882415228247449853, 7.37033755159232893172793045398, 7.72439572473809463411575063494, 9.244533303323341467054082582756

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