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Properties

Label 2-960-80.29-c1-0-3
Degree $2$
Conductor $960$
Sign $-0.894 - 0.447i$
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  + (−0.707 + 0.707i)3-s + (0.860 + 2.06i)5-s + 0.707·7-s − 1.00i·9-s + (−1.79 + 1.79i)11-s + (−3.86 + 3.86i)13-s + (−2.06 − 0.850i)15-s + 0.244i·17-s + (−1.53 − 1.53i)19-s + (−0.500 + 0.500i)21-s + 6.92·23-s + (−3.51 + 3.55i)25-s + (0.707 + 0.707i)27-s + (−4.89 − 4.89i)29-s − 7.60·31-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (0.384 + 0.922i)5-s + 0.267·7-s − 0.333i·9-s + (−0.542 + 0.542i)11-s + (−1.07 + 1.07i)13-s + (−0.533 − 0.219i)15-s + 0.0593i·17-s + (−0.352 − 0.352i)19-s + (−0.109 + 0.109i)21-s + 1.44·23-s + (−0.703 + 0.710i)25-s + (0.136 + 0.136i)27-s + (−0.909 − 0.909i)29-s − 1.36·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(7.66563\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.209146 + 0.886375i\)
\(L(\frac12)\) \(\approx\) \(0.209146 + 0.886375i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (-0.860 - 2.06i)T \)
good7 \( 1 - 0.707T + 7T^{2} \)
11 \( 1 + (1.79 - 1.79i)T - 11iT^{2} \)
13 \( 1 + (3.86 - 3.86i)T - 13iT^{2} \)
17 \( 1 - 0.244iT - 17T^{2} \)
19 \( 1 + (1.53 + 1.53i)T + 19iT^{2} \)
23 \( 1 - 6.92T + 23T^{2} \)
29 \( 1 + (4.89 + 4.89i)T + 29iT^{2} \)
31 \( 1 + 7.60T + 31T^{2} \)
37 \( 1 + (-8.47 - 8.47i)T + 37iT^{2} \)
41 \( 1 - 2.12iT - 41T^{2} \)
43 \( 1 + (0.684 + 0.684i)T + 43iT^{2} \)
47 \( 1 + 4.47iT - 47T^{2} \)
53 \( 1 + (1.47 + 1.47i)T + 53iT^{2} \)
59 \( 1 + (5.86 - 5.86i)T - 59iT^{2} \)
61 \( 1 + (-0.0537 - 0.0537i)T + 61iT^{2} \)
67 \( 1 + (7.85 - 7.85i)T - 67iT^{2} \)
71 \( 1 - 2.08iT - 71T^{2} \)
73 \( 1 + 9.69T + 73T^{2} \)
79 \( 1 - 7.34T + 79T^{2} \)
83 \( 1 + (6.80 - 6.80i)T - 83iT^{2} \)
89 \( 1 + 3.07iT - 89T^{2} \)
97 \( 1 - 1.39iT - 97T^{2} \)
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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

−10.30307973330186645202317725304, −9.701510483696975728458093808666, −8.983356451335029128697328765237, −7.58816142817258668855603223235, −7.03127947931257341834027939637, −6.12918876734461746214447323323, −5.08560961514071541160211440800, −4.33591890469056621658637025896, −2.98144185394602320901574535939, −1.95735842051329751525873830676, 0.42375103400051508239134159372, 1.82084953026687081426009504255, 3.10896961105943417540885973483, 4.65673019251678994437642268292, 5.36234984127394736548328435875, 5.96685646115315836219971187595, 7.34079258951751263623298670938, 7.86227796722854176097757025337, 8.883815265224690239183866222994, 9.570611508492603903024786247658

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