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Properties

Label 1-276-276.71-r0-0-0
Degree $1$
Conductor $276$
Sign $0.947 + 0.320i$
Analytic cond. $1.28173$
Root an. cond. $1.28173$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(276\)    =    \(2^{2} \cdot 3 \cdot 23\)
Sign: $0.947 + 0.320i$
Analytic conductor: \(1.28173\)
Root analytic conductor: \(1.28173\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{276} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 276,\ (0:\ ),\ 0.947 + 0.320i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.094541031 + 0.1800541857i\)
\(L(\frac12)\) \(\approx\) \(1.094541031 + 0.1800541857i\)
\(L(1)\) \(\approx\) \(0.9969408009 + 0.05496970166i\)
\(L(1)\) \(\approx\) \(0.9969408009 + 0.05496970166i\)

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

−25.60834098468160185956756562100, −24.785626940094406122831889597104, −23.569852477395679004787969000077, −22.88760474286672346074885234717, −22.397260232999932494399729178308, −20.77837581556864079185278288948, −20.194669036745527860756918178583, −19.34505230761238196765834250616, −18.27124162486253187903447555628, −17.39988124388022208223548433363, −16.37526234461068349615382906744, −15.36768224564914153239460389623, −14.539661144661680226673528914614, −13.62781498552899714179356763698, −12.31963132306511989333129212668, −11.580035482313874017810605772204, −10.398293216072545201918872382372, −9.748958570734400453230525449675, −7.97315443650240425642494822307, −7.52723953404332524561082875024, −6.429394885987225284127388890839, −4.87318347546717304875245479010, −3.885864727162770980770977689363, −2.82363975554291004086052450259, −0.96484859475625915257388248183, 1.21897622401766449980095004354, 2.846587015660149490388735954262, 4.05111307995109990085229852161, 5.16607721834664191780089216242, 6.26527658354773491653203734544, 7.57354675707778895029319615561, 8.6778221381935333572430645899, 9.18359318491879325519138861693, 10.7992182262212917783034159912, 11.85223458736855577297603361146, 12.26345751449251996020737949723, 13.636337989884562456167878375167, 14.65240762006767462608495516344, 15.66699532763439555909137232068, 16.361048030202404389544182955966, 17.3394553452789632916976147452, 18.691560568184501137924181369772, 19.21852664494366251614112440680, 20.1406999762285293775742947063, 21.408928478552969860864257090608, 21.849386442426290617924940421241, 23.14331814245712771813379405060, 24.08760232059249199968728961015, 24.553259124197874442245668959395, 25.690835794351422040240285431973

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