Properties

Label 2-960-20.19-c2-0-46
Degree $2$
Conductor $960$
Sign $i$
Analytic cond. $26.1581$
Root an. cond. $5.11449$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73·3-s − 5i·5-s + 10.3·7-s + 2.99·9-s − 10.3i·11-s − 18i·13-s − 8.66i·15-s − 10i·17-s + 13.8i·19-s + 18·21-s − 6.92·23-s − 25·25-s + 5.19·27-s − 36·29-s − 6.92i·31-s + ⋯
L(s)  = 1  + 0.577·3-s i·5-s + 1.48·7-s + 0.333·9-s − 0.944i·11-s − 1.38i·13-s − 0.577i·15-s − 0.588i·17-s + 0.729i·19-s + 0.857·21-s − 0.301·23-s − 25-s + 0.192·27-s − 1.24·29-s − 0.223i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $i$
Analytic conductor: \(26.1581\)
Root analytic conductor: \(5.11449\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1),\ i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.615380562\)
\(L(\frac12)\) \(\approx\) \(2.615380562\)
\(L(2)\) 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 \( 1 - 1.73T \)
5 \( 1 + 5iT \)
good7 \( 1 - 10.3T + 49T^{2} \)
11 \( 1 + 10.3iT - 121T^{2} \)
13 \( 1 + 18iT - 169T^{2} \)
17 \( 1 + 10iT - 289T^{2} \)
19 \( 1 - 13.8iT - 361T^{2} \)
23 \( 1 + 6.92T + 529T^{2} \)
29 \( 1 + 36T + 841T^{2} \)
31 \( 1 + 6.92iT - 961T^{2} \)
37 \( 1 - 54iT - 1.36e3T^{2} \)
41 \( 1 - 18T + 1.68e3T^{2} \)
43 \( 1 - 20.7T + 1.84e3T^{2} \)
47 \( 1 + 2.20e3T^{2} \)
53 \( 1 - 26iT - 2.80e3T^{2} \)
59 \( 1 + 31.1iT - 3.48e3T^{2} \)
61 \( 1 - 74T + 3.72e3T^{2} \)
67 \( 1 + 41.5T + 4.48e3T^{2} \)
71 \( 1 + 103. iT - 5.04e3T^{2} \)
73 \( 1 - 36iT - 5.32e3T^{2} \)
79 \( 1 + 90.0iT - 6.24e3T^{2} \)
83 \( 1 + 90.0T + 6.88e3T^{2} \)
89 \( 1 - 18T + 7.92e3T^{2} \)
97 \( 1 - 72iT - 9.40e3T^{2} \)
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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

−9.472345509248548992517221041577, −8.570535907640040429690992742639, −8.068937985410637363649364335998, −7.55737236823692193984324905845, −5.88177630943575908524729801212, −5.23686464897032305053045865667, −4.33392738028215068882207461963, −3.22160525526512845722656753553, −1.84753715509971178583210598063, −0.78196805635026846277212599965, 1.76423393508129699424519992716, 2.33010697967007716358992504207, 3.88193388633757651489564683164, 4.51550869231076721634850722952, 5.72957608371297792952821657852, 7.02296948193864081245242545617, 7.36412226168658085235076617001, 8.343458226474973262960203147887, 9.194374056996591074299170949409, 9.992338277248906737192648095612

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