Properties

Label 2-960-192.155-c1-0-101
Degree $2$
Conductor $960$
Sign $-0.206 + 0.978i$
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  + (0.854 − 1.12i)2-s + (−0.250 + 1.71i)3-s + (−0.540 − 1.92i)4-s + (0.980 + 0.195i)5-s + (1.71 + 1.74i)6-s + (−1.89 − 0.783i)7-s + (−2.63 − 1.03i)8-s + (−2.87 − 0.857i)9-s + (1.05 − 0.938i)10-s + (4.01 − 2.68i)11-s + (3.43 − 0.444i)12-s + (−0.112 − 0.564i)13-s + (−2.49 + 1.46i)14-s + (−0.579 + 1.63i)15-s + (−3.41 + 2.08i)16-s + (−1.39 − 1.39i)17-s + ⋯
L(s)  = 1  + (0.604 − 0.796i)2-s + (−0.144 + 0.989i)3-s + (−0.270 − 0.962i)4-s + (0.438 + 0.0872i)5-s + (0.701 + 0.712i)6-s + (−0.715 − 0.296i)7-s + (−0.930 − 0.366i)8-s + (−0.958 − 0.285i)9-s + (0.334 − 0.296i)10-s + (1.21 − 0.808i)11-s + (0.991 − 0.128i)12-s + (−0.0311 − 0.156i)13-s + (−0.667 + 0.390i)14-s + (−0.149 + 0.421i)15-s + (−0.854 + 0.520i)16-s + (−0.339 − 0.339i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.206 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.206 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $-0.206 + 0.978i$
Analytic conductor: \(7.66563\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (731, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1/2),\ -0.206 + 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09431 - 1.34957i\)
\(L(\frac12)\) \(\approx\) \(1.09431 - 1.34957i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.854 + 1.12i)T \)
3 \( 1 + (0.250 - 1.71i)T \)
5 \( 1 + (-0.980 - 0.195i)T \)
good7 \( 1 + (1.89 + 0.783i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (-4.01 + 2.68i)T + (4.20 - 10.1i)T^{2} \)
13 \( 1 + (0.112 + 0.564i)T + (-12.0 + 4.97i)T^{2} \)
17 \( 1 + (1.39 + 1.39i)T + 17iT^{2} \)
19 \( 1 + (1.02 + 5.13i)T + (-17.5 + 7.27i)T^{2} \)
23 \( 1 + (-4.71 + 1.95i)T + (16.2 - 16.2i)T^{2} \)
29 \( 1 + (-3.25 + 4.86i)T + (-11.0 - 26.7i)T^{2} \)
31 \( 1 + 1.45T + 31T^{2} \)
37 \( 1 + (2.52 + 0.501i)T + (34.1 + 14.1i)T^{2} \)
41 \( 1 + (-0.992 - 2.39i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (2.21 - 1.48i)T + (16.4 - 39.7i)T^{2} \)
47 \( 1 + (-4.25 + 4.25i)T - 47iT^{2} \)
53 \( 1 + (-6.74 - 10.0i)T + (-20.2 + 48.9i)T^{2} \)
59 \( 1 + (-1.94 + 9.78i)T + (-54.5 - 22.5i)T^{2} \)
61 \( 1 + (-0.301 + 0.451i)T + (-23.3 - 56.3i)T^{2} \)
67 \( 1 + (7.14 + 4.77i)T + (25.6 + 61.8i)T^{2} \)
71 \( 1 + (-4.83 + 11.6i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (-1.79 - 4.33i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (5.42 - 5.42i)T - 79iT^{2} \)
83 \( 1 + (11.0 - 2.20i)T + (76.6 - 31.7i)T^{2} \)
89 \( 1 + (3.42 - 8.26i)T + (-62.9 - 62.9i)T^{2} \)
97 \( 1 - 13.5iT - 97T^{2} \)
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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

−9.888641084882977417081227954765, −9.207589478108292378108728370331, −8.714401553194267508429917658503, −6.75319745721125727716950339582, −6.21776974873053398016385788822, −5.19855789492230222008978832250, −4.32065914307475559495546960886, −3.43692680803047951692865681115, −2.61117986602951233667990514607, −0.68831902676712578778741889894, 1.66093791392268379650966165307, 2.97968162025917364916470538748, 4.12009342607855653188590773450, 5.35696030622546217322662557919, 6.11967256471721901547370670763, 6.81060607477169221100460573410, 7.33380965959840095330036158192, 8.598607194411081920918794224624, 9.042368220793457056107381329170, 10.12853021637212515347276392579

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