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Properties

Label 2-960-80.67-c1-0-4
Degree $2$
Conductor $960$
Sign $0.975 - 0.221i$
Analytic cond. $7.66563$
Root an. cond. $2.76868$
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  − i·3-s + (−2.09 − 0.778i)5-s + (2.44 + 2.44i)7-s − 9-s + (0.181 + 0.181i)11-s − 4.59·13-s + (−0.778 + 2.09i)15-s + (4.20 + 4.20i)17-s + (4.75 + 4.75i)19-s + (2.44 − 2.44i)21-s + (5.75 − 5.75i)23-s + (3.78 + 3.26i)25-s + i·27-s + (0.851 − 0.851i)29-s + 4.78i·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + (−0.937 − 0.348i)5-s + (0.922 + 0.922i)7-s − 0.333·9-s + (0.0548 + 0.0548i)11-s − 1.27·13-s + (−0.201 + 0.541i)15-s + (1.01 + 1.01i)17-s + (1.09 + 1.09i)19-s + (0.532 − 0.532i)21-s + (1.19 − 1.19i)23-s + (0.757 + 0.652i)25-s + 0.192i·27-s + (0.158 − 0.158i)29-s + 0.859i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.221i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 - 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.975 - 0.221i$
Analytic conductor: \(7.66563\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (367, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1/2),\ 0.975 - 0.221i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39771 + 0.157104i\)
\(L(\frac12)\) \(\approx\) \(1.39771 + 0.157104i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + iT \)
5 \( 1 + (2.09 + 0.778i)T \)
good7 \( 1 + (-2.44 - 2.44i)T + 7iT^{2} \)
11 \( 1 + (-0.181 - 0.181i)T + 11iT^{2} \)
13 \( 1 + 4.59T + 13T^{2} \)
17 \( 1 + (-4.20 - 4.20i)T + 17iT^{2} \)
19 \( 1 + (-4.75 - 4.75i)T + 19iT^{2} \)
23 \( 1 + (-5.75 + 5.75i)T - 23iT^{2} \)
29 \( 1 + (-0.851 + 0.851i)T - 29iT^{2} \)
31 \( 1 - 4.78iT - 31T^{2} \)
37 \( 1 + 3.97T + 37T^{2} \)
41 \( 1 + 7.99iT - 41T^{2} \)
43 \( 1 - 4.44T + 43T^{2} \)
47 \( 1 + (2.46 - 2.46i)T - 47iT^{2} \)
53 \( 1 - 4.94iT - 53T^{2} \)
59 \( 1 + (-5.89 + 5.89i)T - 59iT^{2} \)
61 \( 1 + (-7.88 - 7.88i)T + 61iT^{2} \)
67 \( 1 - 7.54T + 67T^{2} \)
71 \( 1 - 6.34T + 71T^{2} \)
73 \( 1 + (-4.78 - 4.78i)T + 73iT^{2} \)
79 \( 1 + 3.83T + 79T^{2} \)
83 \( 1 - 6.25iT - 83T^{2} \)
89 \( 1 - 5.02T + 89T^{2} \)
97 \( 1 + (2.96 + 2.96i)T + 97iT^{2} \)
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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

−10.08969571432657437320663719911, −8.945572502397031100326573504023, −8.271038939048416129697745828559, −7.68622773139970004332343853631, −6.86745058711602617317987423893, −5.49056197767004820528888389991, −5.01469697257086132299828972797, −3.71846798616174927556394376983, −2.49814943272535918940390614675, −1.19037111452371387320833097193, 0.810353313546239469429448211973, 2.81278034165844107436029002160, 3.68720066841548925740616995669, 4.84795088800470110285256299132, 5.16851517408398361130809695863, 7.03363410585475957430316809588, 7.40861063326192240212136774622, 8.133198810908852873305213641820, 9.386948422291565653265957866963, 9.921137726770265588401272865736

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