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Properties

Label 2-1960-7.4-c1-0-38
Degree $2$
Conductor $1960$
Sign $-0.991 + 0.126i$
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  + (1.68 − 2.92i)3-s + (0.5 + 0.866i)5-s + (−4.18 − 7.25i)9-s + (−0.313 + 0.543i)11-s − 1.37·13-s + 3.37·15-s + (−2.68 + 4.65i)17-s + (−3.37 − 5.84i)19-s + (−3.37 − 5.84i)23-s + (−0.499 + 0.866i)25-s − 18.1·27-s + 1.37·29-s + (4 − 6.92i)31-s + (1.05 + 1.83i)33-s + (1 + 1.73i)37-s + ⋯
L(s)  = 1  + (0.973 − 1.68i)3-s + (0.223 + 0.387i)5-s + (−1.39 − 2.41i)9-s + (−0.0946 + 0.163i)11-s − 0.380·13-s + 0.870·15-s + (−0.651 + 1.12i)17-s + (−0.773 − 1.34i)19-s + (−0.703 − 1.21i)23-s + (−0.0999 + 0.173i)25-s − 3.48·27-s + 0.254·29-s + (0.718 − 1.24i)31-s + (0.184 + 0.319i)33-s + (0.164 + 0.284i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1960\)    =    \(2^{3} \cdot 5 \cdot 7^{2}\)
Sign: $-0.991 + 0.126i$
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: \((2,\ 1960,\ (\ :1/2),\ -0.991 + 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.681577638\)
\(L(\frac12)\) \(\approx\) \(1.681577638\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.68 + 2.92i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (0.313 - 0.543i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.37T + 13T^{2} \)
17 \( 1 + (2.68 - 4.65i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.37 + 5.84i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.37 + 5.84i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.37T + 29T^{2} \)
31 \( 1 + (-4 + 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1 - 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 4.74T + 41T^{2} \)
43 \( 1 - 2.74T + 43T^{2} \)
47 \( 1 + (5.05 + 8.76i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.372 + 0.644i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4 - 6.92i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.37 + 7.57i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + (-3 + 5.19i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.05 - 1.83i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 13.4T + 83T^{2} \)
89 \( 1 + (1.62 + 2.81i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 18.8T + 97T^{2} \)
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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

−8.550300462199186947811608565905, −8.074398247053050899541948089186, −7.23782814544997371808093649167, −6.42635390320469833423670983345, −6.23373760007879460888811108947, −4.63304864222192474172565493344, −3.48009759212399328790557296931, −2.36906392741702027741345667879, −2.05388001953233852715683069534, −0.48095394240061577358413906220, 1.97050123391877081624312103863, 2.97350550328332118497727542090, 3.78812735484311601744112288211, 4.64263065434443275130608608210, 5.19468840123823791003764299604, 6.17393457721935811324938319983, 7.56641493910949909053868529914, 8.249459532186658967363286655737, 8.902149014927705200265691152204, 9.589743153290684883461433782816

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