↑

Properties

Label 1-6037-6037.4-r0-0-0
Degree $1$
Conductor $6037$
Sign $0.999 + 0.0194i$
Analytic cond. $28.0357$
Root an. cond. $28.0357$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.363 − 0.931i)2-s + (0.748 − 0.662i)3-s + (−0.736 + 0.676i)4-s + (−0.998 + 0.0499i)5-s + (−0.889 − 0.457i)6-s + (−0.280 − 0.959i)7-s + (0.897 + 0.440i)8-s + (0.121 − 0.992i)9-s + (0.409 + 0.912i)10-s + (0.386 − 0.922i)11-s + (−0.102 + 0.994i)12-s + (−0.316 + 0.948i)13-s + (−0.792 + 0.609i)14-s + (−0.714 + 0.699i)15-s + (0.0842 − 0.996i)16-s + (0.911 − 0.412i)17-s + ⋯
L(s)  = 1  + (−0.363 − 0.931i)2-s + (0.748 − 0.662i)3-s + (−0.736 + 0.676i)4-s + (−0.998 + 0.0499i)5-s + (−0.889 − 0.457i)6-s + (−0.280 − 0.959i)7-s + (0.897 + 0.440i)8-s + (0.121 − 0.992i)9-s + (0.409 + 0.912i)10-s + (0.386 − 0.922i)11-s + (−0.102 + 0.994i)12-s + (−0.316 + 0.948i)13-s + (−0.792 + 0.609i)14-s + (−0.714 + 0.699i)15-s + (0.0842 − 0.996i)16-s + (0.911 − 0.412i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6037 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0194i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6037 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0194i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6037\)
Sign: $0.999 + 0.0194i$
Analytic conductor: \(28.0357\)
Root analytic conductor: \(28.0357\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6037} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6037,\ (0:\ ),\ 0.999 + 0.0194i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4388622568 + 0.004263699673i\)
\(L(\frac12)\) \(\approx\) \(0.4388622568 + 0.004263699673i\)
\(L(1)\) \(\approx\) \(0.5680555717 - 0.5041799473i\)
\(L(1)\) \(\approx\) \(0.5680555717 - 0.5041799473i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad6037 \( 1 \)
good2 \( 1 + (-0.363 - 0.931i)T \)
3 \( 1 + (0.748 - 0.662i)T \)
5 \( 1 + (-0.998 + 0.0499i)T \)
7 \( 1 + (-0.280 - 0.959i)T \)
11 \( 1 + (0.386 - 0.922i)T \)
13 \( 1 + (-0.316 + 0.948i)T \)
17 \( 1 + (0.911 - 0.412i)T \)
19 \( 1 + (-0.207 - 0.978i)T \)
23 \( 1 + (-0.895 + 0.445i)T \)
29 \( 1 + (-0.897 + 0.440i)T \)
31 \( 1 + (-0.788 + 0.614i)T \)
37 \( 1 + (0.710 + 0.703i)T \)
41 \( 1 + (0.765 - 0.643i)T \)
43 \( 1 + (-0.607 + 0.794i)T \)
47 \( 1 + (0.0218 - 0.999i)T \)
53 \( 1 + (0.310 + 0.950i)T \)
59 \( 1 + (0.256 + 0.966i)T \)
61 \( 1 + (0.777 + 0.629i)T \)
67 \( 1 + (0.0156 - 0.999i)T \)
71 \( 1 + (-0.102 + 0.994i)T \)
73 \( 1 + (0.0779 + 0.996i)T \)
79 \( 1 + (-0.811 - 0.584i)T \)
83 \( 1 + (-0.102 + 0.994i)T \)
89 \( 1 + (0.923 + 0.383i)T \)
97 \( 1 + (-0.540 + 0.841i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.67332928843713371204098660128, −16.72006814917643752210077091700, −16.31398465878354538927843725708, −15.67859043388570343573882356486, −15.07981931821514002258707608496, −14.65048155639005603896410956414, −14.39186084743462782390941246511, −12.95293185459455022500075460405, −12.72229986970735932075448119157, −11.83458882844143337587659345134, −10.87310450821361793921741087559, −10.037912674547133896464649817, −9.67186791929107498531937320927, −8.96254350529423110913520183438, −8.11789707795485766249263930711, −7.90288546362572207935372113471, −7.25477152897546715612893513775, −6.141665943765098483628667077337, −5.53036748549150221566870369816, −4.80390258629769614716879733440, −3.95191736156068120027704664326, −3.58023917503380241217659246510, −2.43668923413469927394374041186, −1.62245456488802724037449384974, −0.139307848491144031194627200, 0.88108132531007090497835197452, 1.40841100732940470134342630771, 2.53723461437167317239857688412, 3.170199475348978233316980315449, 3.908084257084750031279702786462, 4.13042092681851688082088947341, 5.323009500009027376279815705169, 6.61549955410756297698169805885, 7.22064651843389312990340322830, 7.6921906664659445549509689018, 8.40687865287862331062149345509, 9.07995430686576009669346229079, 9.59438067850015111688323172846, 10.48341743139868359252879729722, 11.285118267175490457401195111514, 11.71140878525279513004409030904, 12.332353039252564085550879113902, 13.12789200934465237246648962434, 13.628293530940759918740669543633, 14.28576610747517223638949556661, 14.75269939237155079012815931799, 15.9541473793281446770483131093, 16.53956223906891515687910790807, 17.03300445927094585568356740478, 18.0215704661685414339660349186

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