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Properties

Label 2-1960-1960.1219-c0-0-0
Degree $2$
Conductor $1960$
Sign $-0.0960 - 0.995i$
Analytic cond. $0.978167$
Root an. cond. $0.989023$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.623 − 0.781i)2-s + (−0.222 − 0.974i)4-s + (−0.900 − 0.433i)5-s + (−0.900 + 0.433i)7-s + (−0.900 − 0.433i)8-s + (0.623 + 0.781i)9-s + (−0.900 + 0.433i)10-s + (−1.12 + 1.40i)11-s + (−0.277 + 0.347i)13-s + (−0.222 + 0.974i)14-s + (−0.900 + 0.433i)16-s + 0.999·18-s − 1.80·19-s + (−0.222 + 0.974i)20-s + (0.400 + 1.75i)22-s + (−0.445 − 1.94i)23-s + ⋯
L(s)  = 1  + (0.623 − 0.781i)2-s + (−0.222 − 0.974i)4-s + (−0.900 − 0.433i)5-s + (−0.900 + 0.433i)7-s + (−0.900 − 0.433i)8-s + (0.623 + 0.781i)9-s + (−0.900 + 0.433i)10-s + (−1.12 + 1.40i)11-s + (−0.277 + 0.347i)13-s + (−0.222 + 0.974i)14-s + (−0.900 + 0.433i)16-s + 0.999·18-s − 1.80·19-s + (−0.222 + 0.974i)20-s + (0.400 + 1.75i)22-s + (−0.445 − 1.94i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0960 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0960 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1960\)    =    \(2^{3} \cdot 5 \cdot 7^{2}\)
Sign: $-0.0960 - 0.995i$
Analytic conductor: \(0.978167\)
Root analytic conductor: \(0.989023\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1960} (1219, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1960,\ (\ :0),\ -0.0960 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2232626696\)
\(L(\frac12)\) \(\approx\) \(0.2232626696\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.623 + 0.781i)T \)
5 \( 1 + (0.900 + 0.433i)T \)
7 \( 1 + (0.900 - 0.433i)T \)
good3 \( 1 + (-0.623 - 0.781i)T^{2} \)
11 \( 1 + (1.12 - 1.40i)T + (-0.222 - 0.974i)T^{2} \)
13 \( 1 + (0.277 - 0.347i)T + (-0.222 - 0.974i)T^{2} \)
17 \( 1 + (0.900 + 0.433i)T^{2} \)
19 \( 1 + 1.80T + T^{2} \)
23 \( 1 + (0.445 + 1.94i)T + (-0.900 + 0.433i)T^{2} \)
29 \( 1 + (0.900 + 0.433i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.277 - 1.21i)T + (-0.900 - 0.433i)T^{2} \)
41 \( 1 + (1.12 + 0.541i)T + (0.623 + 0.781i)T^{2} \)
43 \( 1 + (-0.623 + 0.781i)T^{2} \)
47 \( 1 + (0.277 - 0.347i)T + (-0.222 - 0.974i)T^{2} \)
53 \( 1 + (-0.0990 - 0.433i)T + (-0.900 + 0.433i)T^{2} \)
59 \( 1 + (-1.62 + 0.781i)T + (0.623 - 0.781i)T^{2} \)
61 \( 1 + (0.900 + 0.433i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.900 - 0.433i)T^{2} \)
73 \( 1 + (0.222 - 0.974i)T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (0.222 - 0.974i)T^{2} \)
89 \( 1 + (0.277 + 0.347i)T + (-0.222 + 0.974i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−10.01410890647228522441366147623, −8.802922565232613463183418859752, −8.184154623665657879449044589379, −7.04332416550755718738020800557, −6.46042135151819344056871905253, −5.12612640857742627704720437110, −4.61362456806342441861276086128, −3.97575017144856223661348857749, −2.63565353659575861215068762368, −2.00827906379789942483476286414, 0.12081611833263204135568461770, 2.73811077298135108391935544738, 3.64142525878134383578138715055, 3.94057090520882080092262911909, 5.25851442847699359588534889841, 6.09652335695292297928191464045, 6.78359887515977474621862534457, 7.46885036709127074006180261349, 8.167390106040482456383270690745, 8.879529725283149162106238058107

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