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Properties

Label 2-960-15.14-c0-0-2
Degree $2$
Conductor $960$
Sign $0.707 + 0.707i$
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 + i·5-s − 1.41i·7-s − 1.00i·9-s + (0.707 + 0.707i)15-s + (−1.00 − 1.00i)21-s + 1.41·23-s − 25-s + (−0.707 − 0.707i)27-s + 1.41·35-s + 2i·41-s + 1.41i·43-s + 1.00·45-s − 1.41·47-s − 1.00·49-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)3-s + i·5-s − 1.41i·7-s − 1.00i·9-s + (0.707 + 0.707i)15-s + (−1.00 − 1.00i)21-s + 1.41·23-s − 25-s + (−0.707 − 0.707i)27-s + 1.41·35-s + 2i·41-s + 1.41i·43-s + 1.00·45-s − 1.41·47-s − 1.00·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(0.479102\)
Root analytic conductor: \(0.692172\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :0),\ 0.707 + 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.258232467\)
\(L(\frac12)\) \(\approx\) \(1.258232467\)
\(L(1)\) 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 \( 1 - iT \)
good7 \( 1 + 1.41iT - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - 1.41T + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - 2iT - T^{2} \)
43 \( 1 - 1.41iT - T^{2} \)
47 \( 1 + 1.41T + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + 1.41iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + 1.41T + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T^{2} \)
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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

−10.05559221458814024568136367743, −9.407164531748578763769288639892, −8.176526296393001303838460109806, −7.55801903984939204258646581233, −6.84288064031452528286995467568, −6.28281240089725782717700820834, −4.65276370543786748256698173174, −3.53727090020670864417844174626, −2.83195966371016843776758153539, −1.34317531009062409967794383494, 1.91097938139596868916120632599, 2.97471217989071320800122459540, 4.13704963367706339077909785761, 5.18355593103578933558912311677, 5.60303087869201692278661580958, 7.08962373701661366401641086813, 8.248050013143878694153461153911, 8.802616603375437587928455014691, 9.234841641994241248913082588130, 10.09960419406987210799869225073

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