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Properties

Label 2-1960-1960.939-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: $\chi_{1960} (939, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1960,\ (\ :0),\ -0.0960 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.020404322\)
\(L(\frac12)\) \(\approx\) \(1.020404322\)
\(L(1)\) 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

−8.952870760049401340178230132917, −8.582111539349093076717817616949, −8.130473499335452941705967003132, −6.74994071949094832838761299137, −6.07047307311679674356030962383, −4.91731478239185789958171585095, −4.24166422967205324594730988018, −2.89873416742364831371446548856, −2.11570269596900823231658448660, −0.962838674749607849938140919098, 1.73578701551651777038951832163, 2.17273066953996361157037831003, 4.12712830723887816814381585948, 5.11848596877293764958672371770, 5.43804236362038466154595119283, 6.69460202584725827718857840601, 7.33411900215049399710130291363, 7.82451608307995041914459906274, 8.683891037669131275141867081016, 9.665572066796632492018857127305

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