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Properties

Label 2-960-120.83-c0-0-3
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

\(L(\frac{1}{2})\) \(\approx\) \(1.265225819\)
\(L(\frac12)\) \(\approx\) \(1.265225819\)
\(L(1)\) 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

−9.867361276980374740933399973272, −9.285743428041843358194228791181, −8.470517853961794443203798321152, −7.28609028493941137621051978733, −6.94147762110853728032162804008, −5.91353865190703216446104675350, −4.65746126491685960900442353610, −3.67080008958162466040099891593, −2.43371479769742654387553848338, −1.26502021208251008985161891634, 2.33278710714009158359677376160, 3.00865802169970201021369718166, 3.86430967622701641927142908577, 5.44438898546176689161136295636, 5.95854901342137969004796102304, 6.91883188116262555031127682194, 8.206942128558074288478280330447, 8.849035460556825106620331838431, 9.638969351235205504357373624477, 10.14315286482335506936605779815

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