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Properties

Label 1-4729-4729.1050-r0-0-0
Degree $1$
Conductor $4729$
Sign $0.476 + 0.879i$
Analytic cond. $21.9613$
Root an. cond. $21.9613$
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  + (−0.290 + 0.956i)2-s + (−0.0398 − 0.999i)3-s + (−0.830 − 0.556i)4-s + (0.998 − 0.0478i)5-s + (0.967 + 0.252i)6-s + (−0.721 + 0.692i)7-s + (0.773 − 0.633i)8-s + (−0.996 + 0.0796i)9-s + (−0.244 + 0.969i)10-s + (−0.876 − 0.481i)11-s + (−0.522 + 0.852i)12-s + (−0.753 + 0.657i)13-s + (−0.453 − 0.891i)14-s + (−0.0875 − 0.996i)15-s + (0.380 + 0.924i)16-s + (0.973 − 0.229i)17-s + ⋯
L(s)  = 1  + (−0.290 + 0.956i)2-s + (−0.0398 − 0.999i)3-s + (−0.830 − 0.556i)4-s + (0.998 − 0.0478i)5-s + (0.967 + 0.252i)6-s + (−0.721 + 0.692i)7-s + (0.773 − 0.633i)8-s + (−0.996 + 0.0796i)9-s + (−0.244 + 0.969i)10-s + (−0.876 − 0.481i)11-s + (−0.522 + 0.852i)12-s + (−0.753 + 0.657i)13-s + (−0.453 − 0.891i)14-s + (−0.0875 − 0.996i)15-s + (0.380 + 0.924i)16-s + (0.973 − 0.229i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.476 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.476 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $0.476 + 0.879i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (1050, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ 0.476 + 0.879i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.010091016 + 0.6012132675i\)
\(L(\frac12)\) \(\approx\) \(1.010091016 + 0.6012132675i\)
\(L(1)\) \(\approx\) \(0.8438683049 + 0.1694403638i\)
\(L(1)\) \(\approx\) \(0.8438683049 + 0.1694403638i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (-0.290 + 0.956i)T \)
3 \( 1 + (-0.0398 - 0.999i)T \)
5 \( 1 + (0.998 - 0.0478i)T \)
7 \( 1 + (-0.721 + 0.692i)T \)
11 \( 1 + (-0.876 - 0.481i)T \)
13 \( 1 + (-0.753 + 0.657i)T \)
17 \( 1 + (0.973 - 0.229i)T \)
19 \( 1 + (0.576 - 0.817i)T \)
23 \( 1 + (0.999 - 0.00797i)T \)
29 \( 1 + (0.956 - 0.290i)T \)
31 \( 1 + (0.158 + 0.987i)T \)
37 \( 1 + (-0.305 + 0.952i)T \)
41 \( 1 + (-0.402 - 0.915i)T \)
43 \( 1 + (0.681 + 0.732i)T \)
47 \( 1 + (-0.704 + 0.709i)T \)
53 \( 1 + (0.835 - 0.549i)T \)
59 \( 1 + (-0.351 + 0.936i)T \)
61 \( 1 + (-0.986 - 0.166i)T \)
67 \( 1 + (-0.971 + 0.236i)T \)
71 \( 1 + (-0.997 - 0.0637i)T \)
73 \( 1 + (0.366 - 0.930i)T \)
79 \( 1 + iT \)
83 \( 1 + (-0.275 - 0.961i)T \)
89 \( 1 + (-0.905 + 0.424i)T \)
97 \( 1 + (0.721 - 0.692i)T \)
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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

−18.006432897926961239467035135957, −17.349217335686909284132712406189, −16.80436847726979168741550825565, −16.36564141814653762163817544179, −15.338038322849781932003243523021, −14.55766483541939324961361408887, −13.95064037071520811583928450932, −13.25056803888777980935929809140, −12.60934466538766768085276862575, −12.01300650652503913304763023325, −10.87520092571851540524638037590, −10.4076835003424040524466876316, −9.93141680352233325236196237610, −9.648211969725376983415390946647, −8.78926053712489608756387642195, −7.87306925028834926767798762022, −7.21510824478730971670524564171, −5.95075128189278064604270396947, −5.30911213434608106643112330553, −4.72956906508960291833868990027, −3.81050001876841841690169928119, −2.97698467906675760666481399344, −2.697859935131499606787845719747, −1.5134514930135447080271615341, −0.47105109954979371417783147599, 0.78384043652219830578336330914, 1.58395099261758427176023452475, 2.74480400638322244194522649919, 3.03198344859006695753920462382, 4.85338753649240878009208067099, 5.25182536942524896874018755761, 5.943629302001583256308648685803, 6.57447168247624698807890332528, 7.12247621305719005548226940283, 7.84710124083590136455386267310, 8.803500492777028513866336342206, 9.09714962627128921095867497044, 9.95434551797331754918231944058, 10.56059281397152135766584958732, 11.71364817479969292805320063797, 12.449185069282847986922727074866, 13.08833943530590025363687463979, 13.66719052271016072344202793924, 14.12492445651357084022000533237, 14.88462159108435979286920274426, 15.69597722121832358768651766596, 16.4744966883901831093812701677, 16.88574077608641437996053819975, 17.78434468043008454438826908915, 18.04960391459084563990737280929

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