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Properties

Label 1-1723-1723.59-r1-0-0
Degree $1$
Conductor $1723$
Sign $0.283 - 0.958i$
Analytic cond. $185.162$
Root an. cond. $185.162$
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  + (0.881 + 0.472i)2-s + (0.201 + 0.979i)3-s + (0.552 + 0.833i)4-s + (0.712 + 0.701i)5-s + (−0.286 + 0.958i)6-s + (0.286 + 0.958i)7-s + (0.0929 + 0.995i)8-s + (−0.919 + 0.394i)9-s + (0.296 + 0.955i)10-s + (−0.222 − 0.974i)11-s + (−0.705 + 0.709i)12-s + (−0.750 − 0.661i)13-s + (−0.201 + 0.979i)14-s + (−0.543 + 0.839i)15-s + (−0.388 + 0.921i)16-s + (−0.254 − 0.967i)17-s + ⋯
L(s)  = 1  + (0.881 + 0.472i)2-s + (0.201 + 0.979i)3-s + (0.552 + 0.833i)4-s + (0.712 + 0.701i)5-s + (−0.286 + 0.958i)6-s + (0.286 + 0.958i)7-s + (0.0929 + 0.995i)8-s + (−0.919 + 0.394i)9-s + (0.296 + 0.955i)10-s + (−0.222 − 0.974i)11-s + (−0.705 + 0.709i)12-s + (−0.750 − 0.661i)13-s + (−0.201 + 0.979i)14-s + (−0.543 + 0.839i)15-s + (−0.388 + 0.921i)16-s + (−0.254 − 0.967i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1723 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.283 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1723 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.283 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1723\)
Sign: $0.283 - 0.958i$
Analytic conductor: \(185.162\)
Root analytic conductor: \(185.162\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1723} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1723,\ (1:\ ),\ 0.283 - 0.958i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-1.278263573 + 0.9549134379i\)
\(L(\frac12)\) \(\approx\) \(-1.278263573 + 0.9549134379i\)
\(L(1)\) \(\approx\) \(1.014564158 + 1.314714774i\)
\(L(1)\) \(\approx\) \(1.014564158 + 1.314714774i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1723 \( 1 \)
good2 \( 1 + (0.881 + 0.472i)T \)
3 \( 1 + (0.201 + 0.979i)T \)
5 \( 1 + (0.712 + 0.701i)T \)
7 \( 1 + (0.286 + 0.958i)T \)
11 \( 1 + (-0.222 - 0.974i)T \)
13 \( 1 + (-0.750 - 0.661i)T \)
17 \( 1 + (-0.254 - 0.967i)T \)
19 \( 1 + (-0.989 + 0.141i)T \)
23 \( 1 + (-0.757 - 0.652i)T \)
29 \( 1 + (0.243 + 0.969i)T \)
31 \( 1 + (0.448 + 0.893i)T \)
37 \( 1 + (0.125 + 0.992i)T \)
41 \( 1 + T \)
43 \( 1 + (-0.632 + 0.774i)T \)
47 \( 1 + (0.865 + 0.501i)T \)
53 \( 1 + (0.0710 - 0.997i)T \)
59 \( 1 + (-0.515 - 0.856i)T \)
61 \( 1 + (-0.992 + 0.120i)T \)
67 \( 1 + (0.211 - 0.977i)T \)
71 \( 1 + (-0.988 - 0.152i)T \)
73 \( 1 + (0.0492 - 0.998i)T \)
79 \( 1 + (-0.953 - 0.301i)T \)
83 \( 1 + (0.632 - 0.774i)T \)
89 \( 1 + (-0.859 + 0.511i)T \)
97 \( 1 + (0.811 + 0.584i)T \)
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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

−19.800317066904726340466840064663, −19.07864480377876254947902195121, −18.03518956340814226705391655515, −17.1836814771024907807493514943, −16.943066990755694440513412567412, −15.55398932963509133603024541820, −14.72998712223236623962459763444, −14.05264398169144495942682479252, −13.46303593917332573267000378772, −12.85071755076172325831421747982, −12.2751633673926310473240042213, −11.52697834375015957077590665053, −10.47419230936180846550926718437, −9.84706178767911511891904782257, −8.92519916564539043851813273515, −7.78242322278997299234941342674, −7.12194053626306235051219847573, −6.24526242029308720829390343863, −5.60123508121538661649407655742, −4.373503829706563342109445922915, −4.12607003003199833615804193673, −2.437519236770167400432927601070, −2.028963212637479516279304945355, −1.25147814988911488139843231771, −0.16738485581550303747354984846, 2.06498429312374661238556042284, 2.888651894511248301518081868835, 3.17228074157387546301562149308, 4.60068562500434372960139067731, 5.1126878416962191721512435733, 5.9422118259870167846592126759, 6.47069350597811202940563751688, 7.746566745753929886162239249271, 8.49542457000631974565505672837, 9.2229165459091782949262174989, 10.30998625198206319167077836601, 10.89636260952602958832035312200, 11.69414283953416329190458240962, 12.55191892700361922458480571029, 13.51292991936218044963939549294, 14.301782750890334170484408032461, 14.634585330144903911594001141164, 15.418159748711019730332633742549, 16.03329633580234132169309217096, 16.774296268170648671853461431330, 17.63148317546576724721528112213, 18.29134508244447958747064265837, 19.34772320591290608274531643490, 20.26714006013482176584466057959, 21.11907540671971211614629598429

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