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Properties

Label 2-1960-7.2-c1-0-22
Degree $2$
Conductor $1960$
Sign $0.991 + 0.126i$
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  + (0.5 + 0.866i)3-s + (0.5 − 0.866i)5-s + (1 − 1.73i)9-s + (2.5 + 4.33i)11-s + 13-s + 0.999·15-s + (−1.5 − 2.59i)17-s + (3 − 5.19i)19-s + (3 − 5.19i)23-s + (−0.499 − 0.866i)25-s + 5·27-s − 9·29-s + (−2.5 + 4.33i)33-s + (−3 + 5.19i)37-s + (0.5 + 0.866i)39-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (0.223 − 0.387i)5-s + (0.333 − 0.577i)9-s + (0.753 + 1.30i)11-s + 0.277·13-s + 0.258·15-s + (−0.363 − 0.630i)17-s + (0.688 − 1.19i)19-s + (0.625 − 1.08i)23-s + (−0.0999 − 0.173i)25-s + 0.962·27-s − 1.67·29-s + (−0.435 + 0.753i)33-s + (−0.493 + 0.854i)37-s + (0.0800 + 0.138i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1960\)    =    \(2^{3} \cdot 5 \cdot 7^{2}\)
Sign: $0.991 + 0.126i$
Analytic conductor: \(15.6506\)
Root analytic conductor: \(3.95609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1960} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1960,\ (\ :1/2),\ 0.991 + 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.244747388\)
\(L(\frac12)\) \(\approx\) \(2.244747388\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (-2.5 - 4.33i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - T + 13T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3 + 5.19i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 9T + 29T^{2} \)
31 \( 1 + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3 - 5.19i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 8T + 41T^{2} \)
43 \( 1 - 6T + 43T^{2} \)
47 \( 1 + (1.5 - 2.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6 - 10.3i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4 + 6.92i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2 + 3.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + (5 + 8.66i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.5 + 2.59i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + (-8 + 13.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 7T + 97T^{2} \)
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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

−9.332553416270889585380140866466, −8.730429690271349799535524116944, −7.41999813706334974201587923759, −6.94107811943086167766425063250, −6.01440024702353780238742076011, −4.78159568389824264671923419274, −4.43087035701751567749177872531, −3.35755205614459514150070420827, −2.24228621689730686981176088121, −0.946546546705800330138507346632, 1.20251199677059935968055747400, 2.12522521275519962293858435337, 3.38915812405772355340204699497, 3.97700302886884856563753111998, 5.50626250685651439833212205579, 5.91570817082780188588606260449, 6.98207243731320425248236119915, 7.58450022652915483588733051832, 8.380317096080285351014342993717, 9.109423435930158577668205860901

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