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Properties

Label 1-51-51.14-r0-0-0
Degree $1$
Conductor $51$
Sign $-0.250 + 0.968i$
Analytic cond. $0.236843$
Root an. cond. $0.236843$
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  + (−0.707 + 0.707i)2-s − i·4-s + (−0.382 + 0.923i)5-s + (0.382 + 0.923i)7-s + (0.707 + 0.707i)8-s + (−0.382 − 0.923i)10-s + (−0.923 + 0.382i)11-s + i·13-s + (−0.923 − 0.382i)14-s − 16-s + (0.707 − 0.707i)19-s + (0.923 + 0.382i)20-s + (0.382 − 0.923i)22-s + (0.923 − 0.382i)23-s + (−0.707 − 0.707i)25-s + (−0.707 − 0.707i)26-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s − i·4-s + (−0.382 + 0.923i)5-s + (0.382 + 0.923i)7-s + (0.707 + 0.707i)8-s + (−0.382 − 0.923i)10-s + (−0.923 + 0.382i)11-s + i·13-s + (−0.923 − 0.382i)14-s − 16-s + (0.707 − 0.707i)19-s + (0.923 + 0.382i)20-s + (0.382 − 0.923i)22-s + (0.923 − 0.382i)23-s + (−0.707 − 0.707i)25-s + (−0.707 − 0.707i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.250 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.250 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(51\)    =    \(3 \cdot 17\)
Sign: $-0.250 + 0.968i$
Analytic conductor: \(0.236843\)
Root analytic conductor: \(0.236843\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{51} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 51,\ (0:\ ),\ -0.250 + 0.968i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3487176989 + 0.4503376370i\)
\(L(\frac12)\) \(\approx\) \(0.3487176989 + 0.4503376370i\)
\(L(1)\) \(\approx\) \(0.5821096631 + 0.3720735960i\)
\(L(1)\) \(\approx\) \(0.5821096631 + 0.3720735960i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
good2 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (-0.382 + 0.923i)T \)
7 \( 1 + (0.382 + 0.923i)T \)
11 \( 1 + (-0.923 + 0.382i)T \)
13 \( 1 + iT \)
19 \( 1 + (0.707 - 0.707i)T \)
23 \( 1 + (0.923 - 0.382i)T \)
29 \( 1 + (0.382 - 0.923i)T \)
31 \( 1 + (0.923 + 0.382i)T \)
37 \( 1 + (-0.923 - 0.382i)T \)
41 \( 1 + (-0.382 - 0.923i)T \)
43 \( 1 + (0.707 + 0.707i)T \)
47 \( 1 - iT \)
53 \( 1 + (-0.707 + 0.707i)T \)
59 \( 1 + (0.707 + 0.707i)T \)
61 \( 1 + (-0.382 - 0.923i)T \)
67 \( 1 - T \)
71 \( 1 + (0.923 + 0.382i)T \)
73 \( 1 + (0.382 - 0.923i)T \)
79 \( 1 + (0.923 - 0.382i)T \)
83 \( 1 + (0.707 - 0.707i)T \)
89 \( 1 + iT \)
97 \( 1 + (-0.382 + 0.923i)T \)
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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

−33.2917218498289545077319689749, −31.87095286796649799028132809403, −30.86015943358658581092070656748, −29.60688276743718212368102084100, −28.72021613988489578217580866959, −27.460107438737707432656478451048, −26.82053851538766347997233156298, −25.37382753463453210989827647294, −24.054659226692736655024736352929, −22.7653152352404922272834817042, −20.99701757352353894735505341524, −20.45860145327121745056431909019, −19.31724262598293300683550568196, −17.891910588448563708619229295, −16.845754741772993078604126168431, −15.75384995701636748183645784323, −13.56271302542844602062312897441, −12.53509059919243128637275876547, −11.139490186036275200919058219319, −10.013867674897660504148192356811, −8.40730978345332806263355833385, −7.560271331603845070055476568044, −4.96109339476848830311540389588, −3.33845173899537514750891767435, −1.066742744479837654450623983892, 2.41343799325580077348400638510, 4.97372839961407677084839451686, 6.55725617267046725924655601393, 7.74174834031963718682987293107, 9.10086344163079124834218948123, 10.55184644990613514467821181997, 11.762234982426351976374710148376, 13.91918882164235028581699825101, 15.13704727181052617682611301337, 15.856309462689832709149228361970, 17.57304573319019169554881673355, 18.53420398285512585051388627777, 19.31846523532039700668667527424, 21.07697303308940941569647057240, 22.58438731750797862514636474496, 23.69576394667260440034419907927, 24.828255352093815907977731029264, 26.093403946947787668283599108015, 26.77581645382985823926459859094, 28.1093438216335210924040391851, 28.96694680721640709134414421984, 30.72937361293122377115882178644, 31.60042921951923746307155500024, 33.19391825776703959062519170937, 34.18856898433043915178174174600

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