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Properties

Label 1-51-51.2-r1-0-0
Degree $1$
Conductor $51$
Sign $-0.998 + 0.0465i$
Analytic cond. $5.48071$
Root an. cond. $5.48071$
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  − i·2-s − 4-s + (0.707 − 0.707i)5-s + (−0.707 − 0.707i)7-s + i·8-s + (−0.707 − 0.707i)10-s + (−0.707 − 0.707i)11-s − 13-s + (−0.707 + 0.707i)14-s + 16-s + i·19-s + (−0.707 + 0.707i)20-s + (−0.707 + 0.707i)22-s + (−0.707 − 0.707i)23-s − i·25-s + i·26-s + ⋯
L(s)  = 1  − i·2-s − 4-s + (0.707 − 0.707i)5-s + (−0.707 − 0.707i)7-s + i·8-s + (−0.707 − 0.707i)10-s + (−0.707 − 0.707i)11-s − 13-s + (−0.707 + 0.707i)14-s + 16-s + i·19-s + (−0.707 + 0.707i)20-s + (−0.707 + 0.707i)22-s + (−0.707 − 0.707i)23-s − i·25-s + i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.998 + 0.0465i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.998 + 0.0465i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(51\)    =    \(3 \cdot 17\)
Sign: $-0.998 + 0.0465i$
Analytic conductor: \(5.48071\)
Root analytic conductor: \(5.48071\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{51} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 51,\ (1:\ ),\ -0.998 + 0.0465i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02302884431 - 0.9896645645i\)
\(L(\frac12)\) \(\approx\) \(0.02302884431 - 0.9896645645i\)
\(L(1)\) \(\approx\) \(0.5925173425 - 0.6503920497i\)
\(L(1)\) \(\approx\) \(0.5925173425 - 0.6503920497i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
good2 \( 1 - iT \)
5 \( 1 + (0.707 - 0.707i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
11 \( 1 + (-0.707 - 0.707i)T \)
13 \( 1 - T \)
19 \( 1 + iT \)
23 \( 1 + (-0.707 - 0.707i)T \)
29 \( 1 + (0.707 - 0.707i)T \)
31 \( 1 + (0.707 - 0.707i)T \)
37 \( 1 + (0.707 - 0.707i)T \)
41 \( 1 + (0.707 + 0.707i)T \)
43 \( 1 - iT \)
47 \( 1 + T \)
53 \( 1 - iT \)
59 \( 1 + iT \)
61 \( 1 + (-0.707 - 0.707i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.707 + 0.707i)T \)
73 \( 1 + (-0.707 + 0.707i)T \)
79 \( 1 + (0.707 + 0.707i)T \)
83 \( 1 - iT \)
89 \( 1 + T \)
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

−34.04784741945525695523022861511, −32.86168047098269501867761165076, −31.840979828676910483160680825330, −30.74243710223362401262122147467, −29.22446376383010879353145064174, −28.08316826921019740723404911838, −26.573622992611084673984196974708, −25.75980348021679468933203244156, −24.93703123018965468704997961812, −23.52122292268362702848812300888, −22.30503639446707412884214562685, −21.61867791759719052645361949915, −19.470618145575773585715868045815, −18.22172907780401786757006331170, −17.41696439933236752639251963942, −15.84174818756594733128327397170, −14.930178140953442875724069704302, −13.656268305989007198037692775139, −12.44169838768633531622564169527, −10.182067343168754057968974021620, −9.238386568029485534904120908684, −7.4440483181739074149100217333, −6.29697733053697461875341065917, −4.99879525547512101527739317690, −2.74573570283124026880265507201, 0.560656348539303105788978617580, 2.50670389499911364794201130682, 4.27693333747464259725788792485, 5.81564977993107877714449946964, 8.10893364214305493709549982982, 9.64011025553872299527667341078, 10.423663252825610686600457377145, 12.196786681686833992006974727038, 13.20249564347370590840733592065, 14.163670537860925838273132636079, 16.37239219776283592502667052129, 17.3940341629849484403098168, 18.79725703263568690964219712102, 19.96875749646952332430813109562, 20.9414784630429538242686965597, 21.99665976458978875229489899537, 23.22650241483615758346768671814, 24.53695222733646442118264787044, 26.1417885786690683367843881237, 27.08403482655871180266936656829, 28.59996338299064701999068621019, 29.20991019852028493389281768236, 30.10760518126745989784517051990, 31.837621881382426494162297101794, 32.23079530163812667142659223924

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