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Properties

Label 2-1960-1960.379-c0-0-0
Degree $2$
Conductor $1960$
Sign $0.967 + 0.253i$
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  + (−0.222 − 0.974i)2-s + (−0.900 + 0.433i)4-s + (0.623 + 0.781i)5-s + (0.623 − 0.781i)7-s + (0.623 + 0.781i)8-s + (−0.222 + 0.974i)9-s + (0.623 − 0.781i)10-s + (−0.277 − 1.21i)11-s + (0.400 + 1.75i)13-s + (−0.900 − 0.433i)14-s + (0.623 − 0.781i)16-s + 18-s + 1.24·19-s + (−0.900 − 0.433i)20-s + (−1.12 + 0.541i)22-s + (−1.80 + 0.867i)23-s + ⋯
L(s)  = 1  + (−0.222 − 0.974i)2-s + (−0.900 + 0.433i)4-s + (0.623 + 0.781i)5-s + (0.623 − 0.781i)7-s + (0.623 + 0.781i)8-s + (−0.222 + 0.974i)9-s + (0.623 − 0.781i)10-s + (−0.277 − 1.21i)11-s + (0.400 + 1.75i)13-s + (−0.900 − 0.433i)14-s + (0.623 − 0.781i)16-s + 18-s + 1.24·19-s + (−0.900 − 0.433i)20-s + (−1.12 + 0.541i)22-s + (−1.80 + 0.867i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1960\)    =    \(2^{3} \cdot 5 \cdot 7^{2}\)
Sign: $0.967 + 0.253i$
Analytic conductor: \(0.978167\)
Root analytic conductor: \(0.989023\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1960} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1960,\ (\ :0),\ 0.967 + 0.253i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.099534596\)
\(L(\frac12)\) \(\approx\) \(1.099534596\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.222 + 0.974i)T \)
5 \( 1 + (-0.623 - 0.781i)T \)
7 \( 1 + (-0.623 + 0.781i)T \)
good3 \( 1 + (0.222 - 0.974i)T^{2} \)
11 \( 1 + (0.277 + 1.21i)T + (-0.900 + 0.433i)T^{2} \)
13 \( 1 + (-0.400 - 1.75i)T + (-0.900 + 0.433i)T^{2} \)
17 \( 1 + (-0.623 - 0.781i)T^{2} \)
19 \( 1 - 1.24T + T^{2} \)
23 \( 1 + (1.80 - 0.867i)T + (0.623 - 0.781i)T^{2} \)
29 \( 1 + (-0.623 - 0.781i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-0.400 - 0.193i)T + (0.623 + 0.781i)T^{2} \)
41 \( 1 + (0.277 + 0.347i)T + (-0.222 + 0.974i)T^{2} \)
43 \( 1 + (0.222 + 0.974i)T^{2} \)
47 \( 1 + (-0.400 - 1.75i)T + (-0.900 + 0.433i)T^{2} \)
53 \( 1 + (-1.62 + 0.781i)T + (0.623 - 0.781i)T^{2} \)
59 \( 1 + (-0.777 + 0.974i)T + (-0.222 - 0.974i)T^{2} \)
61 \( 1 + (-0.623 - 0.781i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.623 + 0.781i)T^{2} \)
73 \( 1 + (0.900 + 0.433i)T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (0.900 + 0.433i)T^{2} \)
89 \( 1 + (-0.400 + 1.75i)T + (-0.900 - 0.433i)T^{2} \)
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

−9.573723294410306271185565268809, −8.648611662504711047474899610587, −7.88053810697930722894853339635, −7.23291734202862843826190456617, −6.02133259245097067166073888122, −5.23339901333705236057081520024, −4.17011430280327848278025866819, −3.40584802885643368844038308669, −2.27626949169306438527679604191, −1.48254331030438654358886566651, 0.986339148827016825654174742505, 2.36710694949562786073777003312, 3.86679825330639122614690017547, 4.88081292600592455466734665410, 5.61928639719247933778096507033, 5.94458935368948497789193547326, 7.10735228168551871891186057798, 8.038062273446786271531415081287, 8.457248202626378949674505445653, 9.272402876288352391028055505832

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