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Properties

Label 2-960-120.83-c0-0-2
Degree $2$
Conductor $960$
Sign $0.229 + 0.973i$
Analytic cond. $0.479102$
Root an. cond. $0.692172$
Motivic weight $0$
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 + (−0.707 + 0.707i)5-s + (−1 − i)7-s − 1.00i·9-s − 1.41i·11-s − 1.00i·15-s + 1.41·21-s − 1.00i·25-s + (0.707 + 0.707i)27-s − 1.41i·29-s + (1.00 + 1.00i)33-s + 1.41·35-s + (0.707 + 0.707i)45-s + i·49-s + (−1.41 − 1.41i)53-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)3-s + (−0.707 + 0.707i)5-s + (−1 − i)7-s − 1.00i·9-s − 1.41i·11-s − 1.00i·15-s + 1.41·21-s − 1.00i·25-s + (0.707 + 0.707i)27-s − 1.41i·29-s + (1.00 + 1.00i)33-s + 1.41·35-s + (0.707 + 0.707i)45-s + i·49-s + (−1.41 − 1.41i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.229 + 0.973i$
Analytic conductor: \(0.479102\)
Root analytic conductor: \(0.692172\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (863, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :0),\ 0.229 + 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4121616081\)
\(L(\frac12)\) \(\approx\) \(0.4121616081\)
\(L(1)\) 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 \( 1 + (0.707 - 0.707i)T \)
good7 \( 1 + (1 + i)T + iT^{2} \)
11 \( 1 + 1.41iT - T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + 1.41iT - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (1.41 + 1.41i)T + iT^{2} \)
59 \( 1 + 1.41T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (1 + i)T + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (1 - i)T - iT^{2} \)
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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

−10.19048561541531564472396213365, −9.497752166396538145735674479316, −8.376009124828956151724252789596, −7.40413740577371857177523711192, −6.45328345462200126512332886374, −5.98651646807274834141218568921, −4.57888716449978539701770448898, −3.66559386123698103083288515228, −3.12961762722282852526811288830, −0.43406240891863601879244889421, 1.63671808758740279470374265084, 2.97122021121797226514391073316, 4.44035228393569833582695394968, 5.23037029087958850839305820986, 6.15564799367901701608401138345, 7.04614727082797716696462624221, 7.72860544172485611019105759924, 8.788144031158073430955137884716, 9.466643786857875089208930327978, 10.44823527644084265993014830073

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