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Properties

Label 2-960-15.8-c1-0-8
Degree $2$
Conductor $960$
Sign $0.794 - 0.607i$
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  + (0.618 − 1.61i)3-s − 2.23·5-s + (−1 + i)7-s + (−2.23 − 2.00i)9-s + 4.47i·11-s + (3 + 3i)13-s + (−1.38 + 3.61i)15-s + (2.23 + 2.23i)17-s − 2i·19-s + (1 + 2.23i)21-s + (−2.23 + 2.23i)23-s + 5.00·25-s + (−4.61 + 2.38i)27-s + 4.47·29-s + 4·31-s + ⋯
L(s)  = 1  + (0.356 − 0.934i)3-s − 0.999·5-s + (−0.377 + 0.377i)7-s + (−0.745 − 0.666i)9-s + 1.34i·11-s + (0.832 + 0.832i)13-s + (−0.356 + 0.934i)15-s + (0.542 + 0.542i)17-s − 0.458i·19-s + (0.218 + 0.487i)21-s + (−0.466 + 0.466i)23-s + 1.00·25-s + (−0.888 + 0.458i)27-s + 0.830·29-s + 0.718·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 - 0.607i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.794 - 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.794 - 0.607i$
Analytic conductor: \(7.66563\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (833, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1/2),\ 0.794 - 0.607i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14072 + 0.385863i\)
\(L(\frac12)\) \(\approx\) \(1.14072 + 0.385863i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.618 + 1.61i)T \)
5 \( 1 + 2.23T \)
good7 \( 1 + (1 - i)T - 7iT^{2} \)
11 \( 1 - 4.47iT - 11T^{2} \)
13 \( 1 + (-3 - 3i)T + 13iT^{2} \)
17 \( 1 + (-2.23 - 2.23i)T + 17iT^{2} \)
19 \( 1 + 2iT - 19T^{2} \)
23 \( 1 + (2.23 - 2.23i)T - 23iT^{2} \)
29 \( 1 - 4.47T + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + (-3 + 3i)T - 37iT^{2} \)
41 \( 1 - 8.94iT - 41T^{2} \)
43 \( 1 + (-3 - 3i)T + 43iT^{2} \)
47 \( 1 + (6.70 + 6.70i)T + 47iT^{2} \)
53 \( 1 + (2.23 - 2.23i)T - 53iT^{2} \)
59 \( 1 + 8.94T + 59T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 + (-1 + i)T - 67iT^{2} \)
71 \( 1 - 4.47iT - 71T^{2} \)
73 \( 1 + (-1 - i)T + 73iT^{2} \)
79 \( 1 - 6iT - 79T^{2} \)
83 \( 1 + (6.70 - 6.70i)T - 83iT^{2} \)
89 \( 1 - 4.47T + 89T^{2} \)
97 \( 1 + (-9 + 9i)T - 97iT^{2} \)
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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

−9.957997795679852682282419847126, −9.139560119166136652600021240673, −8.280765059514846230449735601012, −7.65308372489570943359819392403, −6.77957754034941056612945805805, −6.14788747801314144587627277009, −4.69698351076062491034953450516, −3.72827001220566881274346905860, −2.63551777097133147067110496024, −1.34231332985726759130732995103, 0.59507906019160650978863238508, 3.02065628922490800224158870246, 3.49147796184537358946725204729, 4.42170407726469852992939011993, 5.51813783879856691579658446804, 6.42680885105880373850851970383, 7.79504653210529528359344458657, 8.270987987101726918871656281561, 8.983490714114111136648138917787, 10.12044790747949962085215358384

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