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Properties

Label 1-276-276.11-r1-0-0
Degree $1$
Conductor $276$
Sign $-0.354 - 0.934i$
Analytic cond. $29.6603$
Root an. cond. $29.6603$
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  + (0.841 − 0.540i)5-s + (−0.142 + 0.989i)7-s + (−0.415 − 0.909i)11-s + (−0.142 − 0.989i)13-s + (−0.654 + 0.755i)17-s + (−0.654 − 0.755i)19-s + (0.415 − 0.909i)25-s + (0.654 − 0.755i)29-s + (0.959 − 0.281i)31-s + (0.415 + 0.909i)35-s + (−0.841 − 0.540i)37-s + (−0.841 + 0.540i)41-s + (−0.959 − 0.281i)43-s + 47-s + (−0.959 − 0.281i)49-s + ⋯
L(s)  = 1  + (0.841 − 0.540i)5-s + (−0.142 + 0.989i)7-s + (−0.415 − 0.909i)11-s + (−0.142 − 0.989i)13-s + (−0.654 + 0.755i)17-s + (−0.654 − 0.755i)19-s + (0.415 − 0.909i)25-s + (0.654 − 0.755i)29-s + (0.959 − 0.281i)31-s + (0.415 + 0.909i)35-s + (−0.841 − 0.540i)37-s + (−0.841 + 0.540i)41-s + (−0.959 − 0.281i)43-s + 47-s + (−0.959 − 0.281i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.354 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.354 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(276\)    =    \(2^{2} \cdot 3 \cdot 23\)
Sign: $-0.354 - 0.934i$
Analytic conductor: \(29.6603\)
Root analytic conductor: \(29.6603\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{276} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 276,\ (1:\ ),\ -0.354 - 0.934i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7899159326 - 1.144836348i\)
\(L(\frac12)\) \(\approx\) \(0.7899159326 - 1.144836348i\)
\(L(1)\) \(\approx\) \(1.028270488 - 0.2426079125i\)
\(L(1)\) \(\approx\) \(1.028270488 - 0.2426079125i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
23 \( 1 \)
good5 \( 1 + (0.841 - 0.540i)T \)
7 \( 1 + (-0.142 + 0.989i)T \)
11 \( 1 + (-0.415 - 0.909i)T \)
13 \( 1 + (-0.142 - 0.989i)T \)
17 \( 1 + (-0.654 + 0.755i)T \)
19 \( 1 + (-0.654 - 0.755i)T \)
29 \( 1 + (0.654 - 0.755i)T \)
31 \( 1 + (0.959 - 0.281i)T \)
37 \( 1 + (-0.841 - 0.540i)T \)
41 \( 1 + (-0.841 + 0.540i)T \)
43 \( 1 + (-0.959 - 0.281i)T \)
47 \( 1 + T \)
53 \( 1 + (-0.142 + 0.989i)T \)
59 \( 1 + (-0.142 - 0.989i)T \)
61 \( 1 + (0.959 - 0.281i)T \)
67 \( 1 + (0.415 - 0.909i)T \)
71 \( 1 + (0.415 - 0.909i)T \)
73 \( 1 + (-0.654 - 0.755i)T \)
79 \( 1 + (-0.142 - 0.989i)T \)
83 \( 1 + (-0.841 - 0.540i)T \)
89 \( 1 + (-0.959 - 0.281i)T \)
97 \( 1 + (-0.841 + 0.540i)T \)
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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

−25.79713205899744868743350264650, −24.98998592412675641126617189408, −23.778017238558978495319444383417, −23.00494635922291066859442038716, −22.162573734888377926255764037474, −21.09773876068215159031438613504, −20.42111612020407556886603808726, −19.29244992853628572500635310130, −18.308428843899038319697253906, −17.44686429814922617129652036953, −16.69399925246236822641924423784, −15.52160672323871403047347635048, −14.339409213211867375973132159471, −13.76384258261997316099366961959, −12.76968395253531887526413247648, −11.52579965218771224926946751590, −10.32144651588900553665332532764, −9.896003741486162813154037835657, −8.5827207898798910892865538629, −7.037697414241385107494514949044, −6.705034763794573218616225480878, −5.14383312528600117912668426364, −4.08928362132194763009903149316, −2.64035526734263063274525748614, −1.51906495534877733537593113092, 0.40433705153475025267534692878, 2.040286199103249249047089092573, 3.02518113377354926570058994685, 4.73793251789057358973631740767, 5.70932000409778704421161378350, 6.43348845454303071608100235668, 8.24482368017235797058575358654, 8.772062501684024964696644969010, 9.943724062405510042877682475248, 10.89360681720177443768084189232, 12.18404551086657607290825733069, 13.04723663600121909622416290163, 13.753707010255109419314115715473, 15.16211586452140286700006870726, 15.77528780010444068289754568934, 17.05077015055884895569749648760, 17.71210098006764443412359088392, 18.73616586444950775801766802730, 19.65876517408586517367164814943, 20.77789665214410162108414460652, 21.651988921421078062867306407217, 22.11004183786114903398546625804, 23.47387357906248590479706675826, 24.508691602855577480285316407852, 25.0197048429856704274571741016

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