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Properties

Label 2-960-960.29-c0-0-1
Degree $2$
Conductor $960$
Sign $0.881 + 0.471i$
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  + (0.831 − 0.555i)2-s + (0.980 + 0.195i)3-s + (0.382 − 0.923i)4-s + (−0.555 + 0.831i)5-s + (0.923 − 0.382i)6-s + (−0.195 − 0.980i)8-s + (0.923 + 0.382i)9-s + i·10-s + (0.555 − 0.831i)12-s + (−0.707 + 0.707i)15-s + (−0.707 − 0.707i)16-s + (−0.785 − 0.785i)17-s + (0.980 − 0.195i)18-s + (−0.324 + 0.216i)19-s + (0.555 + 0.831i)20-s + ⋯
L(s)  = 1  + (0.831 − 0.555i)2-s + (0.980 + 0.195i)3-s + (0.382 − 0.923i)4-s + (−0.555 + 0.831i)5-s + (0.923 − 0.382i)6-s + (−0.195 − 0.980i)8-s + (0.923 + 0.382i)9-s + i·10-s + (0.555 − 0.831i)12-s + (−0.707 + 0.707i)15-s + (−0.707 − 0.707i)16-s + (−0.785 − 0.785i)17-s + (0.980 − 0.195i)18-s + (−0.324 + 0.216i)19-s + (0.555 + 0.831i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.881 + 0.471i$
Analytic conductor: \(0.479102\)
Root analytic conductor: \(0.692172\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :0),\ 0.881 + 0.471i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.931368142\)
\(L(\frac12)\) \(\approx\) \(1.931368142\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.831 + 0.555i)T \)
3 \( 1 + (-0.980 - 0.195i)T \)
5 \( 1 + (0.555 - 0.831i)T \)
good7 \( 1 + (-0.707 + 0.707i)T^{2} \)
11 \( 1 + (0.923 - 0.382i)T^{2} \)
13 \( 1 + (0.382 - 0.923i)T^{2} \)
17 \( 1 + (0.785 + 0.785i)T + iT^{2} \)
19 \( 1 + (0.324 - 0.216i)T + (0.382 - 0.923i)T^{2} \)
23 \( 1 + (0.636 - 1.53i)T + (-0.707 - 0.707i)T^{2} \)
29 \( 1 + (0.923 + 0.382i)T^{2} \)
31 \( 1 + 0.765iT - T^{2} \)
37 \( 1 + (-0.382 - 0.923i)T^{2} \)
41 \( 1 + (0.707 + 0.707i)T^{2} \)
43 \( 1 + (-0.923 + 0.382i)T^{2} \)
47 \( 1 + (1.38 + 1.38i)T + iT^{2} \)
53 \( 1 + (-0.360 - 1.81i)T + (-0.923 + 0.382i)T^{2} \)
59 \( 1 + (0.382 + 0.923i)T^{2} \)
61 \( 1 + (0.382 + 0.0761i)T + (0.923 + 0.382i)T^{2} \)
67 \( 1 + (-0.923 - 0.382i)T^{2} \)
71 \( 1 + (-0.707 + 0.707i)T^{2} \)
73 \( 1 + (-0.707 - 0.707i)T^{2} \)
79 \( 1 + (1 - i)T - iT^{2} \)
83 \( 1 + (1.53 - 1.02i)T + (0.382 - 0.923i)T^{2} \)
89 \( 1 + (0.707 - 0.707i)T^{2} \)
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

−10.14894278023267241963703464378, −9.618605433397757112224124249437, −8.550214918857332764293275922885, −7.47194967303021773685013093792, −6.88651603014399749940267019407, −5.72843317612446050396974113133, −4.48316680099797692439930478307, −3.76323244335078643415260935420, −2.91437936066249308038468534319, −1.96821148694832773967519628178, 1.95826932727521786148354444771, 3.18524972011652620667748398557, 4.22707237828942128348866532560, 4.70799919164744153865842688558, 6.10626049499754863361912870733, 6.92096597311862464548682223320, 7.86433058245447374772100768711, 8.501098243647205887142795090966, 8.963832073643739186544505044283, 10.26101521648648725569081152395

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