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Properties

Label 2-1960-7.2-c1-0-20
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 + (−1.5 − 2.59i)11-s + 13-s − 0.999·15-s + (2.5 + 4.33i)17-s + (3 − 5.19i)19-s + (−0.499 − 0.866i)25-s + 5·27-s − 5·29-s + (−1 − 1.73i)31-s + (1.5 − 2.59i)33-s + (2 − 3.46i)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 + (−0.452 − 0.783i)11-s + 0.277·13-s − 0.258·15-s + (0.606 + 1.05i)17-s + (0.688 − 1.19i)19-s + (−0.0999 − 0.173i)25-s + 0.962·27-s − 0.928·29-s + (−0.179 − 0.311i)31-s + (0.261 − 0.452i)33-s + (0.328 − 0.569i)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

\(L(1)\) \(\approx\) \(1.893007435\)
\(L(\frac12)\) \(\approx\) \(1.893007435\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - T + 13T^{2} \)
17 \( 1 + (-2.5 - 4.33i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3 + 5.19i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 5T + 29T^{2} \)
31 \( 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 10T + 43T^{2} \)
47 \( 1 + (-4.5 + 7.79i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3 - 5.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6 + 10.3i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1 - 1.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-7 - 12.1i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 9T + 97T^{2} \)
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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

−9.198103648441087822674314351144, −8.460417511804081056516317786271, −7.64969920246848774376651998092, −6.84357197925815552601537426998, −5.94851247347703293403558517743, −5.15884348602755845279518344920, −3.91760116853790225245892836322, −3.49939694392419889397332435366, −2.41402585296171777608287487664, −0.797686843580911226208408625383, 1.12684482311128304382752187342, 2.15535862149470916397892179821, 3.22892444234741820734816255097, 4.34864498740088045384677361358, 5.14632780040259010177477502251, 5.94646938761601244713799742981, 7.22313567828003316424632789883, 7.56854256610346181037789012175, 8.203485005443139313918157609603, 9.231895301645432758306455090403

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