Properties

Label 1-4729-4729.1000-r0-0-0
Degree $1$
Conductor $4729$
Sign $0.355 - 0.934i$
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  + (0.467 − 0.884i)2-s + (0.351 − 0.936i)3-s + (−0.563 − 0.826i)4-s + (−0.908 − 0.417i)5-s + (−0.663 − 0.748i)6-s + (0.821 + 0.569i)7-s + (−0.993 + 0.111i)8-s + (−0.753 − 0.657i)9-s + (−0.793 + 0.608i)10-s + (−0.981 − 0.190i)11-s + (−0.971 + 0.236i)12-s + (−0.984 − 0.174i)13-s + (0.887 − 0.460i)14-s + (−0.709 + 0.704i)15-s + (−0.366 + 0.930i)16-s + (−0.872 − 0.488i)17-s + ⋯
L(s)  = 1  + (0.467 − 0.884i)2-s + (0.351 − 0.936i)3-s + (−0.563 − 0.826i)4-s + (−0.908 − 0.417i)5-s + (−0.663 − 0.748i)6-s + (0.821 + 0.569i)7-s + (−0.993 + 0.111i)8-s + (−0.753 − 0.657i)9-s + (−0.793 + 0.608i)10-s + (−0.981 − 0.190i)11-s + (−0.971 + 0.236i)12-s + (−0.984 − 0.174i)13-s + (0.887 − 0.460i)14-s + (−0.709 + 0.704i)15-s + (−0.366 + 0.930i)16-s + (−0.872 − 0.488i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.355 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.355 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $0.355 - 0.934i$
Analytic conductor: \(21.9613\)
Root analytic conductor: \(21.9613\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (1000, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (0:\ ),\ 0.355 - 0.934i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6806156458 - 0.4691975761i\)
\(L(\frac12)\) \(\approx\) \(0.6806156458 - 0.4691975761i\)
\(L(1)\) \(\approx\) \(0.6233299422 - 0.6804402813i\)
\(L(1)\) \(\approx\) \(0.6233299422 - 0.6804402813i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (0.467 - 0.884i)T \)
3 \( 1 + (0.351 - 0.936i)T \)
5 \( 1 + (-0.908 - 0.417i)T \)
7 \( 1 + (0.821 + 0.569i)T \)
11 \( 1 + (-0.981 - 0.190i)T \)
13 \( 1 + (-0.984 - 0.174i)T \)
17 \( 1 + (-0.872 - 0.488i)T \)
19 \( 1 + (-0.687 + 0.726i)T \)
23 \( 1 + (-0.0717 + 0.997i)T \)
29 \( 1 + (-0.467 - 0.884i)T \)
31 \( 1 + (-0.135 - 0.990i)T \)
37 \( 1 + (-0.336 + 0.941i)T \)
41 \( 1 + (0.830 + 0.556i)T \)
43 \( 1 + (0.894 + 0.446i)T \)
47 \( 1 + (0.732 - 0.681i)T \)
53 \( 1 + (0.864 + 0.502i)T \)
59 \( 1 + (-0.0875 - 0.996i)T \)
61 \( 1 + (-0.997 - 0.0637i)T \)
67 \( 1 + (0.549 - 0.835i)T \)
71 \( 1 + (0.839 - 0.543i)T \)
73 \( 1 + (0.229 + 0.973i)T \)
79 \( 1 - T \)
83 \( 1 + (-0.589 - 0.808i)T \)
89 \( 1 + (-0.721 + 0.692i)T \)
97 \( 1 + (0.821 + 0.569i)T \)
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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

−17.99899369009580967139727358165, −17.4850026192840624239745910992, −16.746146581256244069267195413927, −16.11255964164675080279763423692, −15.44717784309435479226190894635, −15.07405950517909194958663867733, −14.3394332189152553744060124141, −14.08614436354697485791202309556, −12.9424121902853427900693280353, −12.42287821234832735774569737893, −11.42474443559606967734831452011, −10.67196924973804685192586471311, −10.424746943793492617815257964096, −9.03318350661624407598198036385, −8.685028739951637055486568055831, −7.86597311225511650213583272392, −7.33633414519474940140074990913, −6.75866422719741965095701585163, −5.5146102754131046299856799923, −4.910299457876962574557765896, −4.240540058191878421090881462536, −3.97453156128467812801046584700, −2.7799966614783495263993737756, −2.33105116327287069108540713827, −0.26284328582188227625011086531, 0.63952342632216678363975492019, 1.70784374064951961260763483689, 2.36086180464772621799138405447, 2.91506565954129497980721105355, 3.95710832583840542701792894373, 4.63249815618045678390306671437, 5.41900378542247958997690601277, 5.98884986411767682465093350347, 7.19115314689703916643395478337, 7.89989633079380292253071260235, 8.31734870126334677744843783219, 9.150587418942712663882054498997, 9.81828287493858387647671534533, 11.13922241101906059072799526308, 11.228091436679041165014560708793, 12.21391451528898386448632532640, 12.40919438274645279151260034274, 13.259026185162437980743118265850, 13.75207694568906524233687087591, 14.6591130796543581045417031272, 15.21016021308126769041509692532, 15.59773838874102185126099135540, 16.944967093219079125909634625630, 17.575032415211567487162418093434, 18.35348558147364509845730913089

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