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Properties

Label 2-1960-7.2-c1-0-1
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  + (−1.64 − 2.84i)3-s + (0.5 − 0.866i)5-s + (−3.91 + 6.77i)9-s + (−2.91 − 5.04i)11-s − 2.75·13-s − 3.28·15-s + (1 + 1.73i)17-s + (−0.378 + 0.654i)19-s + (0.266 − 0.462i)23-s + (−0.499 − 0.866i)25-s + 15.8·27-s − 0.823·29-s + (−1.28 − 2.23i)31-s + (−9.57 + 16.5i)33-s + (−2.37 + 4.11i)37-s + ⋯
L(s)  = 1  + (−0.949 − 1.64i)3-s + (0.223 − 0.387i)5-s + (−1.30 + 2.25i)9-s + (−0.877 − 1.52i)11-s − 0.764·13-s − 0.849·15-s + (0.242 + 0.420i)17-s + (−0.0867 + 0.150i)19-s + (0.0556 − 0.0963i)23-s + (−0.0999 − 0.173i)25-s + 3.05·27-s − 0.152·29-s + (−0.231 − 0.401i)31-s + (−1.66 + 2.88i)33-s + (−0.390 + 0.677i)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

\(L(1)\) \(\approx\) \(0.08805585022\)
\(L(\frac12)\) \(\approx\) \(0.08805585022\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.64 + 2.84i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (2.91 + 5.04i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.75T + 13T^{2} \)
17 \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.378 - 0.654i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.266 + 0.462i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.823T + 29T^{2} \)
31 \( 1 + (1.28 + 2.23i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.37 - 4.11i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 6.06T + 41T^{2} \)
43 \( 1 - 0.710T + 43T^{2} \)
47 \( 1 + (6.44 - 11.1i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.20 - 7.27i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4 - 6.92i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.70 + 8.14i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.93 + 10.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-1.75 - 3.04i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.75 + 8.23i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6.71T + 83T^{2} \)
89 \( 1 + (-0.878 + 1.52i)T + (-44.5 - 77.0i)T^{2} \)
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

−9.095846567952685555872624827914, −8.072257021783150312961754831485, −7.88384615645731245684433041148, −6.83458701446899742387937772900, −6.09207207227914535955259880448, −5.54160183426460455158801901599, −4.82315295033939018869806241642, −3.13088192946485554874049461227, −2.12607407552685349265513086858, −1.03375252554357535624453800407, 0.04076343457114665019624722729, 2.25744647527864239041125376446, 3.38067573015059140838754672362, 4.31028092473841053967369710600, 5.19210381630481120768908654317, 5.36438911572178357879473333025, 6.68536278504053851585723071364, 7.26079190133549925348411702461, 8.511881931273645125713536066936, 9.484594856137208609110023303776

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