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Properties

Label 1-1280-1280.13-r1-0-0
Degree $1$
Conductor $1280$
Sign $0.994 + 0.108i$
Analytic cond. $137.555$
Root an. cond. $137.555$
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  + (0.956 − 0.290i)3-s + (0.831 + 0.555i)7-s + (0.831 − 0.555i)9-s + (−0.881 + 0.471i)11-s + (0.0980 − 0.995i)13-s + (−0.382 + 0.923i)17-s + (−0.773 + 0.634i)19-s + (0.956 + 0.290i)21-s + (−0.980 − 0.195i)23-s + (0.634 − 0.773i)27-s + (0.471 − 0.881i)29-s + (0.707 − 0.707i)31-s + (−0.707 + 0.707i)33-s + (0.773 + 0.634i)37-s + (−0.195 − 0.980i)39-s + ⋯
L(s)  = 1  + (0.956 − 0.290i)3-s + (0.831 + 0.555i)7-s + (0.831 − 0.555i)9-s + (−0.881 + 0.471i)11-s + (0.0980 − 0.995i)13-s + (−0.382 + 0.923i)17-s + (−0.773 + 0.634i)19-s + (0.956 + 0.290i)21-s + (−0.980 − 0.195i)23-s + (0.634 − 0.773i)27-s + (0.471 − 0.881i)29-s + (0.707 − 0.707i)31-s + (−0.707 + 0.707i)33-s + (0.773 + 0.634i)37-s + (−0.195 − 0.980i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.994 + 0.108i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.994 + 0.108i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1280\)    =    \(2^{8} \cdot 5\)
Sign: $0.994 + 0.108i$
Analytic conductor: \(137.555\)
Root analytic conductor: \(137.555\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1280} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1280,\ (1:\ ),\ 0.994 + 0.108i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.479888176 + 0.1900257364i\)
\(L(\frac12)\) \(\approx\) \(3.479888176 + 0.1900257364i\)
\(L(1)\) \(\approx\) \(1.611813007 - 0.03200320627i\)
\(L(1)\) \(\approx\) \(1.611813007 - 0.03200320627i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.956 - 0.290i)T \)
7 \( 1 + (0.831 + 0.555i)T \)
11 \( 1 + (-0.881 + 0.471i)T \)
13 \( 1 + (0.0980 - 0.995i)T \)
17 \( 1 + (-0.382 + 0.923i)T \)
19 \( 1 + (-0.773 + 0.634i)T \)
23 \( 1 + (-0.980 - 0.195i)T \)
29 \( 1 + (0.471 - 0.881i)T \)
31 \( 1 + (0.707 - 0.707i)T \)
37 \( 1 + (0.773 + 0.634i)T \)
41 \( 1 + (-0.980 - 0.195i)T \)
43 \( 1 + (0.956 + 0.290i)T \)
47 \( 1 + (0.923 + 0.382i)T \)
53 \( 1 + (0.881 - 0.471i)T \)
59 \( 1 + (0.995 - 0.0980i)T \)
61 \( 1 + (0.956 - 0.290i)T \)
67 \( 1 + (0.290 + 0.956i)T \)
71 \( 1 + (0.831 + 0.555i)T \)
73 \( 1 + (0.831 - 0.555i)T \)
79 \( 1 + (-0.382 - 0.923i)T \)
83 \( 1 + (0.773 - 0.634i)T \)
89 \( 1 + (0.195 + 0.980i)T \)
97 \( 1 + (0.707 + 0.707i)T \)
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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

−20.87278576670488593396486983963, −20.065005724395793673762224075533, −19.48574193763616720206683500942, −18.496335163659168749467921802834, −17.99157245761209966095089226881, −16.88206706187979146375485343944, −16.06847780653765561540461395857, −15.49370837345069087024992996217, −14.51418052399671979574190899501, −13.85342811326510314008275735469, −13.491467728984887735391099486992, −12.35522470731623174852076750834, −11.28189224953556496635871853971, −10.65522246850489691356589978465, −9.83106052979936005803377341181, −8.819302666752225904594603713306, −8.33988392473950167657177152170, −7.40721063074436109766649629551, −6.73989638223022359992489752924, −5.295091876508333143577981846443, −4.5281694859112339686523896789, −3.83332420475970354811692718571, −2.645820993507127392050917073839, −1.98885358874703563415541756555, −0.72943037887676031230598442691, 0.84460060733212040377477706849, 2.16715088589742127345393940376, 2.42380636085137919941102670909, 3.77718779426084767664609444710, 4.54383542242995590563166002659, 5.65822040166663409020806527655, 6.48274156004261057984709832164, 7.90511654958398595454347223302, 7.98295797767164203281355730016, 8.769848220378857478837204449225, 9.99963729632743219669615225836, 10.42897110394237456598856963676, 11.645967703420500826020787805885, 12.52949081731574108552928891125, 13.07784678916441897671451261347, 13.929905028478996553271037232313, 14.90042205687311227006231926692, 15.20060307991463284784479677555, 15.95094123372507052214738884417, 17.35192611320352072422586758659, 17.84211841068127405964546139904, 18.65426903191506711632216226153, 19.23297519697080022398974188382, 20.292122368341369702026934877321, 20.68451773848973523975414664114

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