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Properties

Label 1-4729-4729.3838-r1-0-0
Degree $1$
Conductor $4729$
Sign $0.311 - 0.950i$
Analytic cond. $508.201$
Root an. cond. $508.201$
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  + (0.868 − 0.495i)2-s + (−0.582 + 0.812i)3-s + (0.509 − 0.860i)4-s + (−0.488 − 0.872i)5-s + (−0.103 + 0.994i)6-s + (0.952 − 0.305i)7-s + (0.0159 − 0.999i)8-s + (−0.321 − 0.947i)9-s + (−0.856 − 0.516i)10-s + (0.139 + 0.990i)11-s + (0.402 + 0.915i)12-s + (−0.198 + 0.980i)13-s + (0.675 − 0.737i)14-s + (0.993 + 0.111i)15-s + (−0.481 − 0.876i)16-s + (0.805 − 0.592i)17-s + ⋯
L(s)  = 1  + (0.868 − 0.495i)2-s + (−0.582 + 0.812i)3-s + (0.509 − 0.860i)4-s + (−0.488 − 0.872i)5-s + (−0.103 + 0.994i)6-s + (0.952 − 0.305i)7-s + (0.0159 − 0.999i)8-s + (−0.321 − 0.947i)9-s + (−0.856 − 0.516i)10-s + (0.139 + 0.990i)11-s + (0.402 + 0.915i)12-s + (−0.198 + 0.980i)13-s + (0.675 − 0.737i)14-s + (0.993 + 0.111i)15-s + (−0.481 − 0.876i)16-s + (0.805 − 0.592i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.311 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.311 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4729\)
Sign: $0.311 - 0.950i$
Analytic conductor: \(508.201\)
Root analytic conductor: \(508.201\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4729} (3838, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4729,\ (1:\ ),\ 0.311 - 0.950i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.441402643 - 1.768617818i\)
\(L(\frac12)\) \(\approx\) \(2.441402643 - 1.768617818i\)
\(L(1)\) \(\approx\) \(1.395711947 - 0.3828770283i\)
\(L(1)\) \(\approx\) \(1.395711947 - 0.3828770283i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4729 \( 1 \)
good2 \( 1 + (0.868 - 0.495i)T \)
3 \( 1 + (-0.582 + 0.812i)T \)
5 \( 1 + (-0.488 - 0.872i)T \)
7 \( 1 + (0.952 - 0.305i)T \)
11 \( 1 + (0.139 + 0.990i)T \)
13 \( 1 + (-0.198 + 0.980i)T \)
17 \( 1 + (0.805 - 0.592i)T \)
19 \( 1 + (-0.709 + 0.704i)T \)
23 \( 1 + (0.940 - 0.339i)T \)
29 \( 1 + (-0.964 + 0.263i)T \)
31 \( 1 + (-0.131 + 0.991i)T \)
37 \( 1 + (-0.984 - 0.174i)T \)
41 \( 1 + (0.0279 - 0.999i)T \)
43 \( 1 + (0.391 + 0.920i)T \)
47 \( 1 + (0.217 - 0.976i)T \)
53 \( 1 + (-0.982 + 0.186i)T \)
59 \( 1 + (0.633 - 0.773i)T \)
61 \( 1 + (0.841 - 0.539i)T \)
67 \( 1 + (-0.830 - 0.556i)T \)
71 \( 1 + (0.358 + 0.933i)T \)
73 \( 1 + (-0.190 - 0.981i)T \)
79 \( 1 + (-0.707 + 0.707i)T \)
83 \( 1 + (0.351 + 0.936i)T \)
89 \( 1 + (0.225 + 0.974i)T \)
97 \( 1 + (0.305 + 0.952i)T \)
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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

−17.98987957151962024013588114687, −17.24520591782794249319358945773, −17.03934123439873982823646644162, −15.96514444500511415427630404100, −15.316785233288331118647013799081, −14.6496722660157513618078268629, −14.29557428595136353937267710253, −13.229243410718997837105497694399, −12.997682600530790700336380416617, −11.92094217056909453519701218955, −11.58836303882660329447654753395, −10.909714455281008959867577538007, −10.50637113579844349848736627533, −8.85297372766429883072114256312, −8.1304046809467326898068267148, −7.65019559155634083106571489945, −7.11706651083821432464650037263, −6.169911236940235817841327366023, −5.75125297933427234769853084724, −5.088311027620925156842144327175, −4.17972343495223618014706551613, −3.24967740106799395817430966661, −2.64457663718621358889050066162, −1.77403442886894677075137670815, −0.65087040779238972017399333665, 0.4580659208976797356928248782, 1.40410434873084424953919663344, 1.98773609858619371036073495751, 3.342775987225613500233921486726, 4.03617045573300461071460937361, 4.53204535857610014159186588751, 5.12122270714049427115079765936, 5.535979765833223115355303557274, 6.785304768888514927468546692638, 7.25725550129898123812812279493, 8.39318114281302404610723834702, 9.24895172651796977756181709135, 9.76137038117563741179786266531, 10.68907851508739568181037364465, 11.12041418734314880199797773989, 11.97155201616188493225992451098, 12.22403475349299021934189326448, 12.89051852401173467731124510490, 14.02808947209243126372462418973, 14.547585574728600333935434087159, 15.03930911434352539905906258301, 15.808455888746759471787249999885, 16.49230149693059490598952488610, 16.966938006530663865584869251451, 17.68576036158982880921307238887

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