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Properties

Label 2-1960-7.2-c1-0-24
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  + (−1.20 − 2.09i)3-s + (0.5 − 0.866i)5-s + (−1.41 + 2.44i)9-s + (2.41 + 4.18i)11-s − 2·13-s − 2.41·15-s + (1.82 + 3.16i)17-s + (2.82 − 4.89i)19-s + (4.20 − 7.28i)23-s + (−0.499 − 0.866i)25-s − 0.414·27-s − 2.17·29-s + (−2.41 − 4.18i)31-s + (5.82 − 10.0i)33-s + (2.82 − 4.89i)37-s + ⋯
L(s)  = 1  + (−0.696 − 1.20i)3-s + (0.223 − 0.387i)5-s + (−0.471 + 0.816i)9-s + (0.727 + 1.26i)11-s − 0.554·13-s − 0.623·15-s + (0.443 + 0.768i)17-s + (0.648 − 1.12i)19-s + (0.877 − 1.51i)23-s + (−0.0999 − 0.173i)25-s − 0.0797·27-s − 0.403·29-s + (−0.433 − 0.751i)31-s + (1.01 − 1.75i)33-s + (0.464 − 0.805i)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

\(L(1)\) \(\approx\) \(1.290865239\)
\(L(\frac12)\) \(\approx\) \(1.290865239\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.20 + 2.09i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (-2.41 - 4.18i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 + (-1.82 - 3.16i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.82 + 4.89i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.20 + 7.28i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.17T + 29T^{2} \)
31 \( 1 + (2.41 + 4.18i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.82 + 4.89i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 0.171T + 41T^{2} \)
43 \( 1 - 12.8T + 43T^{2} \)
47 \( 1 + (-0.171 + 0.297i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.82 + 4.89i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.32 - 4.03i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.44 + 5.97i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + (3.82 + 6.63i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 13.2T + 83T^{2} \)
89 \( 1 + (8.32 - 14.4i)T + (-44.5 - 77.0i)T^{2} \)
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

−9.051370204632113729753371332489, −7.79333479819788476323020999134, −7.27990413293202968331146073729, −6.60358104084779261442531334988, −5.86871739422377481973774961366, −4.94961150681286790626784460892, −4.15025242153619191721745797788, −2.55735138254886223937107574474, −1.65333404675059905987554940209, −0.58286117478084295658072423903, 1.22475454935689919639932097588, 3.02685226922706435411782475494, 3.62953137543969708399175099860, 4.63130382762962622319403544108, 5.64655965797221910996436744012, 5.82144114676232867457707226665, 7.09450385044435699208095370670, 7.81980798297970723852048772522, 9.124720072635858419929371112607, 9.424232928419045525835078703842

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