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Properties

Label 2-1960-7.2-c1-0-4
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  + (−1 − 1.73i)3-s + (−0.5 + 0.866i)5-s + (−0.499 + 0.866i)9-s + (0.5 + 0.866i)11-s + 3·13-s + 1.99·15-s + (−1 − 1.73i)17-s + (−2.5 + 4.33i)19-s + (−3.5 + 6.06i)23-s + (−0.499 − 0.866i)25-s − 4.00·27-s − 6·29-s + (2 + 3.46i)31-s + (0.999 − 1.73i)33-s + (2.5 − 4.33i)37-s + ⋯
L(s)  = 1  + (−0.577 − 0.999i)3-s + (−0.223 + 0.387i)5-s + (−0.166 + 0.288i)9-s + (0.150 + 0.261i)11-s + 0.832·13-s + 0.516·15-s + (−0.242 − 0.420i)17-s + (−0.573 + 0.993i)19-s + (−0.729 + 1.26i)23-s + (−0.0999 − 0.173i)25-s − 0.769·27-s − 1.11·29-s + (0.359 + 0.622i)31-s + (0.174 − 0.301i)33-s + (0.410 − 0.711i)37-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: $\chi_{1960} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1960,\ (\ :1/2),\ 0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9066367587\)
\(L(\frac12)\) \(\approx\) \(0.9066367587\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 + 1.73i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 3T + 13T^{2} \)
17 \( 1 + (1 + 1.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.5 - 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.5 - 6.06i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.5 + 4.33i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 5T + 41T^{2} \)
43 \( 1 - 6T + 43T^{2} \)
47 \( 1 + (4.5 - 7.79i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.5 + 9.52i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4 - 6.92i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6 - 10.3i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4T + 71T^{2} \)
73 \( 1 + (-6 - 10.3i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (7 - 12.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 6T + 97T^{2} \)
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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

−9.321651539676489981663414754950, −8.306200387651408987341462356947, −7.53689981091878564986593317378, −7.01725939711515055639824937836, −6.04368449582268823240536042515, −5.73002376479802169829029793518, −4.28081560281998341919468480497, −3.50311324456330301325603241294, −2.13006107416386283590282070532, −1.17397570701143918203395662333, 0.38991862446237965611773256504, 2.04064952679526463951370381090, 3.45768349910297191515617908531, 4.33118417454365400517022232495, 4.77631265998901889030915020096, 5.90861127441862910397381835284, 6.38085062405541775631951690032, 7.63102494361054254692186417646, 8.428784275500972841118872782813, 9.134081321630978594977132083773

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