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Properties

Label 1-4303-4303.1020-r0-0-0
Degree $1$
Conductor $4303$
Sign $0.974 - 0.223i$
Analytic cond. $19.9830$
Root an. cond. $19.9830$
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  + (−0.994 − 0.104i)2-s + (−0.449 − 0.893i)3-s + (0.978 + 0.207i)4-s + (0.791 − 0.610i)5-s + (0.353 + 0.935i)6-s + (−0.424 + 0.905i)7-s + (−0.951 − 0.309i)8-s + (−0.595 + 0.803i)9-s + (−0.851 + 0.524i)10-s + (0.524 + 0.851i)11-s + (−0.254 − 0.967i)12-s + (0.516 − 0.856i)14-s + (−0.901 − 0.432i)15-s + (0.913 + 0.406i)16-s + (−0.999 − 0.0380i)17-s + (0.676 − 0.736i)18-s + ⋯
L(s)  = 1  + (−0.994 − 0.104i)2-s + (−0.449 − 0.893i)3-s + (0.978 + 0.207i)4-s + (0.791 − 0.610i)5-s + (0.353 + 0.935i)6-s + (−0.424 + 0.905i)7-s + (−0.951 − 0.309i)8-s + (−0.595 + 0.803i)9-s + (−0.851 + 0.524i)10-s + (0.524 + 0.851i)11-s + (−0.254 − 0.967i)12-s + (0.516 − 0.856i)14-s + (−0.901 − 0.432i)15-s + (0.913 + 0.406i)16-s + (−0.999 − 0.0380i)17-s + (0.676 − 0.736i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4303 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.223i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4303 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.223i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4303\)    =    \(13 \cdot 331\)
Sign: $0.974 - 0.223i$
Analytic conductor: \(19.9830\)
Root analytic conductor: \(19.9830\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4303} (1020, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4303,\ (0:\ ),\ 0.974 - 0.223i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9762492876 - 0.1105736590i\)
\(L(\frac12)\) \(\approx\) \(0.9762492876 - 0.1105736590i\)
\(L(1)\) \(\approx\) \(0.6759574552 - 0.1424231598i\)
\(L(1)\) \(\approx\) \(0.6759574552 - 0.1424231598i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
331 \( 1 \)
good2 \( 1 + (-0.994 - 0.104i)T \)
3 \( 1 + (-0.449 - 0.893i)T \)
5 \( 1 + (0.791 - 0.610i)T \)
7 \( 1 + (-0.424 + 0.905i)T \)
11 \( 1 + (0.524 + 0.851i)T \)
17 \( 1 + (-0.999 - 0.0380i)T \)
19 \( 1 + (0.244 + 0.969i)T \)
23 \( 1 + (0.580 - 0.814i)T \)
29 \( 1 + (0.683 + 0.730i)T \)
31 \( 1 - iT \)
37 \( 1 + (0.999 - 0.00951i)T \)
41 \( 1 + (-0.703 - 0.710i)T \)
43 \( 1 + (0.595 - 0.803i)T \)
47 \( 1 + (0.755 + 0.654i)T \)
53 \( 1 + (-0.254 - 0.967i)T \)
59 \( 1 + (0.556 + 0.830i)T \)
61 \( 1 + (0.786 + 0.618i)T \)
67 \( 1 + (-0.983 + 0.179i)T \)
71 \( 1 + (0.0190 + 0.999i)T \)
73 \( 1 + (0.717 - 0.696i)T \)
79 \( 1 + (-0.959 + 0.281i)T \)
83 \( 1 + (-0.951 - 0.309i)T \)
89 \( 1 + (0.998 + 0.0475i)T \)
97 \( 1 + (0.992 + 0.123i)T \)
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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

−18.12523859860802826403940629495, −17.41958763037748121988908302221, −17.24830843114767381548035461354, −16.46189923047446683071894951808, −15.83107842297605708707865451913, −15.25015104468804668606549968607, −14.42877068246965580458520046828, −13.76283827977538072347265811501, −13.03524487323764008187428615924, −11.76342906279593473709746160782, −11.16983796381207968498763272792, −10.77561011142090594302778104209, −10.12407940835010239012080480270, −9.38050326396745372265611257734, −9.106977328457063220641655231309, −8.10746905258707946547551173226, −6.99309422474658153161408510642, −6.59200576774993024375979483349, −6.0054998685351417474989395767, −5.13763340321000055394819546180, −4.14171267227975304696115147057, −3.1604904428309311964216818159, −2.73344057218110914093922038936, −1.400472512490388174607064755206, −0.57486753447143318889016267374, 0.73980427893549106638389899654, 1.58668853708053384048531216984, 2.25069778517103510206430375569, 2.731041338104143835706290450664, 4.19321690892565597389172000158, 5.28970770122919519176035657837, 5.91525086449294294387922262444, 6.56809296700904973864646837009, 7.07551033685439760572981807378, 8.06950448116593800357641674180, 8.77214064127902818898124880189, 9.19840573710747163473194758458, 10.02653408481388047232671837677, 10.66507803320895565528309663965, 11.68025212610762178652037787826, 12.11530260375682056355516435515, 12.74264147256856900370001875553, 13.20795984520557224169651412184, 14.324435283848217361463959891936, 14.99642818121923535575955934335, 15.997980838193367640724502125508, 16.45838932064185395292373344724, 17.27609308860747757285220327932, 17.57972285111499411700158606838, 18.3527933948354613822631011514

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