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Properties

Label 2-1960-280.269-c0-0-3
Degree $2$
Conductor $1960$
Sign $-0.980 + 0.197i$
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  + (0.866 − 0.5i)2-s + (−1.22 − 0.707i)3-s + (0.499 − 0.866i)4-s + (0.258 − 0.965i)5-s − 1.41·6-s − 0.999i·8-s + (0.499 + 0.866i)9-s + (−0.258 − 0.965i)10-s + (−1.22 + 0.707i)12-s − 1.41i·13-s + (−1 + 0.999i)15-s + (−0.5 − 0.866i)16-s + (0.866 + 0.5i)18-s + (0.707 + 1.22i)19-s + (−0.707 − 0.707i)20-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (−1.22 − 0.707i)3-s + (0.499 − 0.866i)4-s + (0.258 − 0.965i)5-s − 1.41·6-s − 0.999i·8-s + (0.499 + 0.866i)9-s + (−0.258 − 0.965i)10-s + (−1.22 + 0.707i)12-s − 1.41i·13-s + (−1 + 0.999i)15-s + (−0.5 − 0.866i)16-s + (0.866 + 0.5i)18-s + (0.707 + 1.22i)19-s + (−0.707 − 0.707i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 + 0.197i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 + 0.197i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1960\)    =    \(2^{3} \cdot 5 \cdot 7^{2}\)
Sign: $-0.980 + 0.197i$
Analytic conductor: \(0.978167\)
Root analytic conductor: \(0.989023\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1960} (1109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1960,\ (\ :0),\ -0.980 + 0.197i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.252550194\)
\(L(\frac12)\) \(\approx\) \(1.252550194\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 + (-0.258 + 0.965i)T \)
7 \( 1 \)
good3 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + 1.41iT - T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + 1.41iT - T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
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

−9.167522852271841019122355891144, −8.017699192670676004275812348099, −7.26400532373612700068621589586, −6.18974750645661331044211614366, −5.64162841978912579505955899892, −5.24554085391705428808128537292, −4.24350296271399323614233855221, −3.09370678928127672447313815593, −1.69342521978807308097302994280, −0.820788499829898176647567008986, 2.17268057134395361074998855203, 3.31368919161627778270711877609, 4.25539119011858936954506811338, 4.97044417659257465490174410289, 5.69322017542088066845782300717, 6.60961227071775075239537500582, 6.82986973454560003005521521644, 7.86860484482360392541616160318, 9.121715784660266425404794174864, 9.811620690178423773037249365021

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