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Properties

Label 2-1960-7.2-c1-0-30
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  + (−1 − 1.73i)3-s + (−0.5 + 0.866i)5-s + (−0.499 + 0.866i)9-s + (−2 − 3.46i)11-s + 2·13-s + 1.99·15-s + (−1 + 1.73i)19-s + (2 − 3.46i)23-s + (−0.499 − 0.866i)25-s − 4.00·27-s + 10·29-s + (−2 − 3.46i)31-s + (−3.99 + 6.92i)33-s + (1 − 1.73i)37-s + (−2 − 3.46i)39-s + ⋯
L(s)  = 1  + (−0.577 − 0.999i)3-s + (−0.223 + 0.387i)5-s + (−0.166 + 0.288i)9-s + (−0.603 − 1.04i)11-s + 0.554·13-s + 0.516·15-s + (−0.229 + 0.397i)19-s + (0.417 − 0.722i)23-s + (−0.0999 − 0.173i)25-s − 0.769·27-s + 1.85·29-s + (−0.359 − 0.622i)31-s + (−0.696 + 1.20i)33-s + (0.164 − 0.284i)37-s + (−0.320 − 0.554i)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

\(L(1)\) \(\approx\) \(0.5881875967\)
\(L(\frac12)\) \(\approx\) \(0.5881875967\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 + 1.73i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (2 + 3.46i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1 - 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 10T + 29T^{2} \)
31 \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1 + 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 12T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (2 - 3.46i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1 + 1.73i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5 + 8.66i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3 - 5.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2 + 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + (-2 - 3.46i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 14T + 83T^{2} \)
89 \( 1 + (4 - 6.92i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 8T + 97T^{2} \)
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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

−8.394103003709783179871406074336, −8.094931948394374216804053641383, −7.02395400974049047387764823666, −6.43242723721581732518618445923, −5.85114260334268851153971226231, −4.84032235104455141887920943262, −3.62644502470734658448130307549, −2.71905306366609883607589813878, −1.40942627623808003791232793947, −0.23924823840989363834036037935, 1.55012403627239419765466950027, 2.98945410855689800002583134443, 4.04699960680343292029668420016, 4.88099877546702950533352616469, 5.19163953181779158949396316990, 6.36363319755879462400237017109, 7.21131493635429122891472952457, 8.133836720361265078082033428551, 8.894560463961169330410621558776, 9.737164637164440922460775479793

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