L-function 1-51-51.26-r1-0-0

• Label: 1-51-51.26-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: $0$

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 + ⋯

Functional equation

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

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:	$\chi_{51} (26, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 51,\ (1:\ ),\ -0.998 - 0.0465i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.02302884431 + 0.9896645645i$
$L(1)$	$\approx$	$0.5925173425 + 0.6503920497i$

Euler product

   $L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$

	$p$	$F_p(T)$
bad	3	$1$
	17	$1$
good	2	$1 + iT$
	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	$1 + (0.707 + 0.707i)T$

Imaginary part of the first few zeros on the critical line

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

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