Properties

Label 2-1960-7.2-c1-0-21
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.591 − 1.02i)3-s + (0.5 − 0.866i)5-s + (0.799 − 1.38i)9-s + (−0.115 − 0.199i)11-s + 2.27·13-s − 1.18·15-s + (3.26 + 5.65i)17-s + (−0.130 + 0.225i)19-s + (4.43 − 7.68i)23-s + (−0.499 − 0.866i)25-s − 5.44·27-s + 5.42·29-s + (2.43 + 4.21i)31-s + (−0.136 + 0.236i)33-s + (0.577 − 0.999i)37-s + ⋯
L(s)  = 1  + (−0.341 − 0.591i)3-s + (0.223 − 0.387i)5-s + (0.266 − 0.461i)9-s + (−0.0347 − 0.0601i)11-s + 0.630·13-s − 0.305·15-s + (0.792 + 1.37i)17-s + (−0.0298 + 0.0516i)19-s + (0.924 − 1.60i)23-s + (−0.0999 − 0.173i)25-s − 1.04·27-s + 1.00·29-s + (0.437 + 0.757i)31-s + (−0.0237 + 0.0410i)33-s + (0.0948 − 0.164i)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: $\chi_{1960} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1960,\ (\ :1/2),\ 0.198 + 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.745477221\)
\(L(\frac12)\) \(\approx\) \(1.745477221\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.591 + 1.02i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (0.115 + 0.199i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.27T + 13T^{2} \)
17 \( 1 + (-3.26 - 5.65i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.130 - 0.225i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.43 + 7.68i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 5.42T + 29T^{2} \)
31 \( 1 + (-2.43 - 4.21i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.577 + 0.999i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 4.43T + 41T^{2} \)
43 \( 1 - 4.17T + 43T^{2} \)
47 \( 1 + (-0.461 + 0.800i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.06 - 1.84i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.53 + 4.39i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.44 + 7.69i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.07 + 7.06i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 9.06T + 71T^{2} \)
73 \( 1 + (-3.00 - 5.20i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.0564 - 0.0977i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 8.72T + 83T^{2} \)
89 \( 1 + (-7.95 + 13.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 4.29T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.758631180671159822701724503031, −8.393625280226665237702286381433, −7.37176376097000166707654070014, −6.44163969587584561841856689884, −6.09496236183202842708696480499, −5.02218722142924440381091026405, −4.08884268443447051799092862422, −3.07428575773319411775007687217, −1.68704157344683643149480635699, −0.798743030076855565944921064766, 1.19459121441659270554653876078, 2.61446534234353864162439783016, 3.52063331187929457686622257379, 4.57131177148529631874517007788, 5.31143217146428834138792148893, 6.00670968526171716046772538251, 7.13302058280545459373977770228, 7.61059152571488341349713469679, 8.684543673238420126930958801616, 9.574704851256352658212921640258

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