Properties

Label 2-960-80.69-c1-0-19
Degree $2$
Conductor $960$
Sign $0.148 + 0.988i$
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 + (−2.15 + 0.607i)5-s − 2.25·7-s + 1.00i·9-s + (1.66 + 1.66i)11-s + (−4.76 − 4.76i)13-s + (−1.95 − 1.09i)15-s − 6.99i·17-s + (2.66 − 2.66i)19-s + (−1.59 − 1.59i)21-s + 4.41·23-s + (4.26 − 2.61i)25-s + (−0.707 + 0.707i)27-s + (2.59 − 2.59i)29-s + 3.93·31-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (−0.962 + 0.271i)5-s − 0.851·7-s + 0.333i·9-s + (0.500 + 0.500i)11-s + (−1.32 − 1.32i)13-s + (−0.503 − 0.281i)15-s − 1.69i·17-s + (0.611 − 0.611i)19-s + (−0.347 − 0.347i)21-s + 0.921·23-s + (0.852 − 0.522i)25-s + (−0.136 + 0.136i)27-s + (0.481 − 0.481i)29-s + 0.706·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.148 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.148 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.148 + 0.988i$
Analytic conductor: \(7.66563\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1/2),\ 0.148 + 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.646817 - 0.556763i\)
\(L(\frac12)\) \(\approx\) \(0.646817 - 0.556763i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.15 - 0.607i)T \)
good7 \( 1 + 2.25T + 7T^{2} \)
11 \( 1 + (-1.66 - 1.66i)T + 11iT^{2} \)
13 \( 1 + (4.76 + 4.76i)T + 13iT^{2} \)
17 \( 1 + 6.99iT - 17T^{2} \)
19 \( 1 + (-2.66 + 2.66i)T - 19iT^{2} \)
23 \( 1 - 4.41T + 23T^{2} \)
29 \( 1 + (-2.59 + 2.59i)T - 29iT^{2} \)
31 \( 1 - 3.93T + 31T^{2} \)
37 \( 1 + (2.01 - 2.01i)T - 37iT^{2} \)
41 \( 1 + 4.50iT - 41T^{2} \)
43 \( 1 + (7.14 - 7.14i)T - 43iT^{2} \)
47 \( 1 + 10.1iT - 47T^{2} \)
53 \( 1 + (-0.649 + 0.649i)T - 53iT^{2} \)
59 \( 1 + (5.64 + 5.64i)T + 59iT^{2} \)
61 \( 1 + (5.00 - 5.00i)T - 61iT^{2} \)
67 \( 1 + (4.95 + 4.95i)T + 67iT^{2} \)
71 \( 1 - 2.33iT - 71T^{2} \)
73 \( 1 + 2.18T + 73T^{2} \)
79 \( 1 + 6.38T + 79T^{2} \)
83 \( 1 + (-5.25 - 5.25i)T + 83iT^{2} \)
89 \( 1 + 15.7iT - 89T^{2} \)
97 \( 1 - 4.61iT - 97T^{2} \)
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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

−9.795007348646290096932814356490, −9.183143719190538415887477209457, −8.088366194903253040675425941457, −7.27846030193546096975723282824, −6.76654811615188358967545694763, −5.17626154875530169970045036584, −4.56387255266769993882669959189, −3.15329823696255986568138826133, −2.84584264538914130487509930731, −0.39083505947376584940511205694, 1.44038014522899544842868012628, 2.99079615861767214402184201787, 3.82877844181237533039874895309, 4.76910465570823111644719753112, 6.18948340627358738114641369884, 6.88948956037533117713720443995, 7.68567486967081125202687194157, 8.588916493299387777694193279568, 9.217724797251740638931176660600, 10.09120770322655396876203606250

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