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Properties

Label 2-1960-7.2-c1-0-36
Degree $2$
Conductor $1960$
Sign $-0.827 - 0.561i$
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  + (−0.939 − 1.62i)3-s + (0.5 − 0.866i)5-s + (−0.267 + 0.462i)9-s + (1.64 + 2.85i)11-s − 4.19·13-s − 1.87·15-s + (−0.718 − 1.24i)17-s + (0.622 − 1.07i)19-s + (0.136 − 0.236i)23-s + (−0.499 − 0.866i)25-s − 4.63·27-s − 2.36·29-s + (−1.86 − 3.22i)31-s + (3.09 − 5.36i)33-s + (−0.0848 + 0.147i)37-s + ⋯
L(s)  = 1  + (−0.542 − 0.939i)3-s + (0.223 − 0.387i)5-s + (−0.0890 + 0.154i)9-s + (0.496 + 0.860i)11-s − 1.16·13-s − 0.485·15-s + (−0.174 − 0.301i)17-s + (0.142 − 0.247i)19-s + (0.0284 − 0.0492i)23-s + (−0.0999 − 0.173i)25-s − 0.892·27-s − 0.438·29-s + (−0.334 − 0.579i)31-s + (0.539 − 0.933i)33-s + (−0.0139 + 0.0241i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.827 - 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.827 - 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1960\)    =    \(2^{3} \cdot 5 \cdot 7^{2}\)
Sign: $-0.827 - 0.561i$
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: \((2,\ 1960,\ (\ :1/2),\ -0.827 - 0.561i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3471031634\)
\(L(\frac12)\) \(\approx\) \(0.3471031634\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.939 + 1.62i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (-1.64 - 2.85i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 4.19T + 13T^{2} \)
17 \( 1 + (0.718 + 1.24i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.622 + 1.07i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.136 + 0.236i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.36T + 29T^{2} \)
31 \( 1 + (1.86 + 3.22i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.0848 - 0.147i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 11.6T + 41T^{2} \)
43 \( 1 + 10.1T + 43T^{2} \)
47 \( 1 + (-1.56 + 2.70i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.62 + 8.00i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.53 - 7.85i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.63 + 6.29i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.57 - 11.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.87T + 71T^{2} \)
73 \( 1 + (7.62 + 13.2i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.47 - 12.9i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 0.167T + 83T^{2} \)
89 \( 1 + (-1.54 + 2.68i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 6.60T + 97T^{2} \)
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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

−8.683445104529600887679591691815, −7.74173963913439294275061479231, −6.95634456573938419001684146381, −6.63039520651048346092661243287, −5.45184077705348047519040147106, −4.86599634917632413569212395645, −3.76389725070684186598297634500, −2.32430560168335318850680527149, −1.49875803878151169821230430118, −0.12960961069312011111755297147, 1.76561194909211296714125071193, 3.09254395603471887825800316230, 3.91311965297432826899204066448, 4.91111396378007556364144859792, 5.48784957064920211351119704525, 6.40635555549458597275819336680, 7.18657705990805934794168840356, 8.160065047562779364051932874719, 9.031632667852806422175481842465, 9.860915118738651485562201702678

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