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Properties

Label 1-960-960.629-r1-0-0
Degree $1$
Conductor $960$
Sign $-0.0980 + 0.995i$
Analytic cond. $103.166$
Root an. cond. $103.166$
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)7-s + (0.923 + 0.382i)11-s + (0.382 + 0.923i)13-s − i·17-s + (0.382 + 0.923i)19-s + (−0.707 + 0.707i)23-s + (0.923 − 0.382i)29-s − 31-s + (−0.382 + 0.923i)37-s + (0.707 − 0.707i)41-s + (−0.923 − 0.382i)43-s − i·47-s + i·49-s + (−0.923 − 0.382i)53-s + (0.382 − 0.923i)59-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)7-s + (0.923 + 0.382i)11-s + (0.382 + 0.923i)13-s − i·17-s + (0.382 + 0.923i)19-s + (−0.707 + 0.707i)23-s + (0.923 − 0.382i)29-s − 31-s + (−0.382 + 0.923i)37-s + (0.707 − 0.707i)41-s + (−0.923 − 0.382i)43-s − i·47-s + i·49-s + (−0.923 − 0.382i)53-s + (0.382 − 0.923i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0980 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0980 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $-0.0980 + 0.995i$
Analytic conductor: \(103.166\)
Root analytic conductor: \(103.166\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (629, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 960,\ (1:\ ),\ -0.0980 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8598386203 + 0.9486857240i\)
\(L(\frac12)\) \(\approx\) \(0.8598386203 + 0.9486857240i\)
\(L(1)\) \(\approx\) \(0.9749363945 + 0.07559437028i\)
\(L(1)\) \(\approx\) \(0.9749363945 + 0.07559437028i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-0.707 - 0.707i)T \)
11 \( 1 + (0.923 + 0.382i)T \)
13 \( 1 + (0.382 + 0.923i)T \)
17 \( 1 - iT \)
19 \( 1 + (0.382 + 0.923i)T \)
23 \( 1 + (-0.707 + 0.707i)T \)
29 \( 1 + (0.923 - 0.382i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.382 + 0.923i)T \)
41 \( 1 + (0.707 - 0.707i)T \)
43 \( 1 + (-0.923 - 0.382i)T \)
47 \( 1 - iT \)
53 \( 1 + (-0.923 - 0.382i)T \)
59 \( 1 + (0.382 - 0.923i)T \)
61 \( 1 + (0.923 - 0.382i)T \)
67 \( 1 + (-0.923 + 0.382i)T \)
71 \( 1 + (-0.707 - 0.707i)T \)
73 \( 1 + (-0.707 + 0.707i)T \)
79 \( 1 + iT \)
83 \( 1 + (0.382 + 0.923i)T \)
89 \( 1 + (0.707 + 0.707i)T \)
97 \( 1 + T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.65572526621694183217429096210, −20.461525519577589084550644506651, −19.66186146629662941675338160794, −19.20731842298132910030021099366, −18.09756047760050554782936405147, −17.60274321980093858440412550350, −16.42960398856320083230166268861, −15.939714654182390444309947387150, −14.996078144671742353858868132690, −14.31786274479681244803354352218, −13.20204535177049527562073594411, −12.62531312659414472041080613460, −11.786250751715726041980628829356, −10.85487418073493184275573416249, −10.023887903617384865435103293081, −9.01424715613062153214631751742, −8.51570046763665838921446596836, −7.36573573417890767019662351800, −6.24353557841023154450814458242, −5.87122714225426053478718783416, −4.60734302719629712074120024305, −3.519035982907607617449322302876, −2.79236821415948518358783389027, −1.51691523488459920686706546190, −0.30268571836790880811589185286, 1.02302994163043274909916524838, 2.023364849542196364608498020687, 3.459403540017344671264718928246, 3.978424137730922074934226769442, 5.089797900473016361462035958603, 6.29232309063780698999582367709, 6.88684112048248310213234602902, 7.73733372729049172520335500193, 8.922399548934070662478655038600, 9.66288572298057935799355198895, 10.28867178262862296109372450615, 11.54286950397162000080530890001, 11.98492706652063110310399042647, 13.08134453133318277607099186410, 13.961705520439424863514369394088, 14.34123240683383335417375913543, 15.64324564827413044157426987052, 16.29724846968024806201214091584, 16.93402081103831647117824287178, 17.82217514577128734110365570481, 18.73343379604841141428666057334, 19.45763137656374326922091506556, 20.22494195184277679853617736802, 20.83306200675482282986745261679, 21.95364403949956952139347811779

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