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Properties

Label 1-2300-2300.103-r1-0-0
Degree $1$
Conductor $2300$
Sign $-0.338 + 0.940i$
Analytic cond. $247.169$
Root an. cond. $247.169$
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  + (−0.999 + 0.0285i)3-s + (0.989 + 0.142i)7-s + (0.998 − 0.0570i)9-s + (0.198 − 0.980i)11-s + (−0.717 + 0.696i)13-s + (−0.856 + 0.516i)17-s + (0.921 − 0.389i)19-s + (−0.993 − 0.113i)21-s + (−0.996 + 0.0855i)27-s + (0.921 + 0.389i)29-s + (0.0285 − 0.999i)31-s + (−0.170 + 0.985i)33-s + (−0.0570 − 0.998i)37-s + (0.696 − 0.717i)39-s + (−0.362 + 0.931i)41-s + ⋯
L(s)  = 1  + (−0.999 + 0.0285i)3-s + (0.989 + 0.142i)7-s + (0.998 − 0.0570i)9-s + (0.198 − 0.980i)11-s + (−0.717 + 0.696i)13-s + (−0.856 + 0.516i)17-s + (0.921 − 0.389i)19-s + (−0.993 − 0.113i)21-s + (−0.996 + 0.0855i)27-s + (0.921 + 0.389i)29-s + (0.0285 − 0.999i)31-s + (−0.170 + 0.985i)33-s + (−0.0570 − 0.998i)37-s + (0.696 − 0.717i)39-s + (−0.362 + 0.931i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.338 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.338 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2300\)    =    \(2^{2} \cdot 5^{2} \cdot 23\)
Sign: $-0.338 + 0.940i$
Analytic conductor: \(247.169\)
Root analytic conductor: \(247.169\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2300} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2300,\ (1:\ ),\ -0.338 + 0.940i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5154922998 + 0.7333589140i\)
\(L(\frac12)\) \(\approx\) \(0.5154922998 + 0.7333589140i\)
\(L(1)\) \(\approx\) \(0.8141570053 + 0.04390086159i\)
\(L(1)\) \(\approx\) \(0.8141570053 + 0.04390086159i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good3 \( 1 + (-0.999 + 0.0285i)T \)
7 \( 1 + (0.989 + 0.142i)T \)
11 \( 1 + (0.198 - 0.980i)T \)
13 \( 1 + (-0.717 + 0.696i)T \)
17 \( 1 + (-0.856 + 0.516i)T \)
19 \( 1 + (0.921 - 0.389i)T \)
29 \( 1 + (0.921 + 0.389i)T \)
31 \( 1 + (0.0285 - 0.999i)T \)
37 \( 1 + (-0.0570 - 0.998i)T \)
41 \( 1 + (-0.362 + 0.931i)T \)
43 \( 1 + (0.281 - 0.959i)T \)
47 \( 1 + (-0.951 + 0.309i)T \)
53 \( 1 + (0.170 + 0.985i)T \)
59 \( 1 + (0.696 + 0.717i)T \)
61 \( 1 + (-0.941 - 0.336i)T \)
67 \( 1 + (-0.676 + 0.736i)T \)
71 \( 1 + (-0.993 - 0.113i)T \)
73 \( 1 + (-0.996 + 0.0855i)T \)
79 \( 1 + (-0.897 + 0.441i)T \)
83 \( 1 + (0.633 + 0.774i)T \)
89 \( 1 + (0.610 + 0.791i)T \)
97 \( 1 + (-0.633 + 0.774i)T \)
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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

−19.201852531206431403173521045733, −18.085558786723704619715528960215, −17.799730755397533670165202258680, −17.35829812664701928323432821305, −16.432018219162438449542507614497, −15.67526946293523673680768680057, −15.02131754568212433795705283814, −14.24274022830218114129550567323, −13.36116339091963470005717616507, −12.52905388549123981064300803463, −11.81402785857851046513883286654, −11.4457906834441357555091458937, −10.298546495838077749353445593823, −10.07322173987192372204414364737, −8.96346080536190197879351296041, −7.90953690854141239300542723133, −7.28223632035273497410603256487, −6.619071786163008711362466380, −5.57909052861041348712766450752, −4.739833242178833245720138771943, −4.587700133792389142287531685960, −3.190907640185600442890201284539, −2.007533356774563135090046439485, −1.26229750348745901552939275240, −0.2111322343192015726997995546, 0.87138424904196641647679039297, 1.692460899428728408096684128595, 2.72369933265743920790379109096, 4.04221046738718787631941171994, 4.630275539610884424590644012756, 5.422115665403821833001985777400, 6.12409079288004853968419739942, 6.97512078959202769648620908882, 7.69889515739139164592785965547, 8.6737727005517704466664114403, 9.37726413054001264710124304891, 10.388681122840635400092246644812, 11.06083418112799319488003956372, 11.64324804525943415937129212893, 12.11752918533177263156313297220, 13.17191783171323436556765307882, 13.864361519932515130023508323615, 14.6765988891074013137535898997, 15.44846049604551401714326307437, 16.25058501937965635784951150377, 16.82790079198979346315642237610, 17.611681505846053445462928324, 18.02240301267626964064491365267, 18.88062837089464011958234683558, 19.545541440338357387013137186135

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