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Properties

Label 1-275-275.97-r1-0-0
Degree $1$
Conductor $275$
Sign $0.956 + 0.292i$
Analytic cond. $29.5528$
Root an. cond. $29.5528$
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  + (−0.951 + 0.309i)2-s − i·3-s + (0.809 − 0.587i)4-s + (0.309 + 0.951i)6-s + (−0.951 + 0.309i)7-s + (−0.587 + 0.809i)8-s − 9-s + (−0.587 − 0.809i)12-s − i·13-s + (0.809 − 0.587i)14-s + (0.309 − 0.951i)16-s + (−0.951 + 0.309i)17-s + (0.951 − 0.309i)18-s + (0.809 + 0.587i)19-s + (0.309 + 0.951i)21-s + ⋯
L(s)  = 1  + (−0.951 + 0.309i)2-s − i·3-s + (0.809 − 0.587i)4-s + (0.309 + 0.951i)6-s + (−0.951 + 0.309i)7-s + (−0.587 + 0.809i)8-s − 9-s + (−0.587 − 0.809i)12-s − i·13-s + (0.809 − 0.587i)14-s + (0.309 − 0.951i)16-s + (−0.951 + 0.309i)17-s + (0.951 − 0.309i)18-s + (0.809 + 0.587i)19-s + (0.309 + 0.951i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.956 + 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.956 + 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $0.956 + 0.292i$
Analytic conductor: \(29.5528\)
Root analytic conductor: \(29.5528\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 275,\ (1:\ ),\ 0.956 + 0.292i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6998432919 + 0.1047286043i\)
\(L(\frac12)\) \(\approx\) \(0.6998432919 + 0.1047286043i\)
\(L(1)\) \(\approx\) \(0.5851082483 - 0.07527794347i\)
\(L(1)\) \(\approx\) \(0.5851082483 - 0.07527794347i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.951 + 0.309i)T \)
3 \( 1 - iT \)
7 \( 1 + (-0.951 + 0.309i)T \)
13 \( 1 - iT \)
17 \( 1 + (-0.951 + 0.309i)T \)
19 \( 1 + (0.809 + 0.587i)T \)
23 \( 1 + (-0.587 + 0.809i)T \)
29 \( 1 + (0.809 - 0.587i)T \)
31 \( 1 + (-0.809 + 0.587i)T \)
37 \( 1 + (0.587 - 0.809i)T \)
41 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 - iT \)
47 \( 1 + iT \)
53 \( 1 + (-0.951 - 0.309i)T \)
59 \( 1 + (0.809 + 0.587i)T \)
61 \( 1 + T \)
67 \( 1 + (0.951 + 0.309i)T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 + (-0.951 - 0.309i)T \)
79 \( 1 + (-0.309 + 0.951i)T \)
83 \( 1 + (0.951 - 0.309i)T \)
89 \( 1 + (0.809 + 0.587i)T \)
97 \( 1 + (0.951 + 0.309i)T \)
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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

−25.93878642196926008916376389074, −24.812117947477586612435706184595, −23.6139860715525832173848109311, −22.20067759287563576853474424422, −21.87949759239799205290932961194, −20.54989460873969700784354217531, −20.05188209714726680546588031287, −19.14232772537312420338878489865, −18.042764411082412773751736036727, −17.01009969206736471576161465540, −16.16768121501311006032476876424, −15.763430022598978758440385061706, −14.40185927807007497201927489703, −13.164649354564694085633703621234, −11.85317326543652889297536216733, −11.05088189273812012026910730530, −10.048239005168599705998585263719, −9.35841808819330042484320776673, −8.58553395635617004581266275640, −7.11393763251184578905891530091, −6.20684723884510678797150071780, −4.52511676476409929512209876395, −3.43816304858494146174757921151, −2.36450708069864234618720886811, −0.429803673550529659894008085549, 0.76652557202799076186901351053, 2.131235057770083450380131723794, 3.22549555566864262249633994808, 5.56998567840514373448862572730, 6.26259356486561331383163627979, 7.29379479616882453020150928427, 8.14417091533541927350862286014, 9.15278728670871538722558750712, 10.16112679923299839663489596547, 11.32872308967609526813922825505, 12.329701735568091221525086635349, 13.23953535987851314820262544454, 14.418882127798847236912115483178, 15.579846080547095909555857315365, 16.325369846500357689607260696138, 17.60325410445152403235814334544, 18.00622500985789005121270830426, 19.08296110531018490376985426137, 19.72218153010298606502682368057, 20.42624703490381345174741603682, 22.04291360319598198066736444996, 23.041502657780597877204857745, 23.899140777062962635681090751994, 24.97055448185365022952477300696, 25.29827078117227579199352459692

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