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Properties

Label 2-1960-7.4-c1-0-29
Degree $2$
Conductor $1960$
Sign $0.266 + 0.963i$
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  + (−0.780 + 1.35i)3-s + (0.5 + 0.866i)5-s + (0.280 + 0.486i)9-s + (−0.780 + 1.35i)11-s − 6.68·13-s − 1.56·15-s + (3.78 − 6.54i)17-s + (−3.56 − 6.16i)19-s + (−1.56 − 2.70i)23-s + (−0.499 + 0.866i)25-s − 5.56·27-s + 0.438·29-s + (3.12 − 5.40i)31-s + (−1.21 − 2.11i)33-s + (4.12 + 7.14i)37-s + ⋯
L(s)  = 1  + (−0.450 + 0.780i)3-s + (0.223 + 0.387i)5-s + (0.0935 + 0.162i)9-s + (−0.235 + 0.407i)11-s − 1.85·13-s − 0.403·15-s + (0.916 − 1.58i)17-s + (−0.817 − 1.41i)19-s + (−0.325 − 0.563i)23-s + (−0.0999 + 0.173i)25-s − 1.07·27-s + 0.0814·29-s + (0.560 − 0.971i)31-s + (−0.212 − 0.367i)33-s + (0.677 + 1.17i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1960\)    =    \(2^{3} \cdot 5 \cdot 7^{2}\)
Sign: $0.266 + 0.963i$
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: \((2,\ 1960,\ (\ :1/2),\ 0.266 + 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5991270101\)
\(L(\frac12)\) \(\approx\) \(0.5991270101\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.780 - 1.35i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (0.780 - 1.35i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 6.68T + 13T^{2} \)
17 \( 1 + (-3.78 + 6.54i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.56 + 6.16i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.56 + 2.70i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 0.438T + 29T^{2} \)
31 \( 1 + (-3.12 + 5.40i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.12 - 7.14i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 1.12T + 41T^{2} \)
43 \( 1 + 7.12T + 43T^{2} \)
47 \( 1 + (-1.21 - 2.11i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.56 + 11.3i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2 - 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.43 + 5.95i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.12 - 1.94i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (2.12 - 3.67i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.342 + 0.592i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + (-2.56 - 4.43i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 1.31T + 97T^{2} \)
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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

−9.419469685127961928508860226389, −8.169175468317545019837924357476, −7.31351541110583872444616912332, −6.79481640067223442222688769188, −5.59044226854599545434220826296, −4.79214651036639063929924069982, −4.49345930144745702437824573195, −2.91945731425704302135340834025, −2.28211865435842905960583466534, −0.23164004417153024271393660749, 1.26732189887941605994062395590, 2.17273841141671272030388473004, 3.53549871345603245393771640207, 4.49941419469602688282933060173, 5.67848868732071471174292024960, 5.98030808831399409839303244947, 7.02928749724659006174838864824, 7.75841111112105187696537159701, 8.352931285530039682674529143129, 9.416112012252383661715682756961

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