↑

Properties

Label 1-2600-2600.2453-r1-0-0
Degree $1$
Conductor $2600$
Sign $0.191 - 0.981i$
Analytic cond. $279.408$
Root an. cond. $279.408$
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  + (−0.743 − 0.669i)3-s + (0.866 + 0.5i)7-s + (0.104 + 0.994i)9-s + (0.104 − 0.994i)11-s + (0.743 − 0.669i)17-s + (0.669 + 0.743i)19-s + (−0.309 − 0.951i)21-s + (−0.994 − 0.104i)23-s + (0.587 − 0.809i)27-s + (0.669 − 0.743i)29-s + (0.309 − 0.951i)31-s + (−0.743 + 0.669i)33-s + (−0.406 + 0.913i)37-s + (0.913 + 0.406i)41-s + (0.866 + 0.5i)43-s + ⋯
L(s)  = 1  + (−0.743 − 0.669i)3-s + (0.866 + 0.5i)7-s + (0.104 + 0.994i)9-s + (0.104 − 0.994i)11-s + (0.743 − 0.669i)17-s + (0.669 + 0.743i)19-s + (−0.309 − 0.951i)21-s + (−0.994 − 0.104i)23-s + (0.587 − 0.809i)27-s + (0.669 − 0.743i)29-s + (0.309 − 0.951i)31-s + (−0.743 + 0.669i)33-s + (−0.406 + 0.913i)37-s + (0.913 + 0.406i)41-s + (0.866 + 0.5i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.191 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.191 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2600\)    =    \(2^{3} \cdot 5^{2} \cdot 13\)
Sign: $0.191 - 0.981i$
Analytic conductor: \(279.408\)
Root analytic conductor: \(279.408\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2600} (2453, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2600,\ (1:\ ),\ 0.191 - 0.981i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.595726622 - 1.314652573i\)
\(L(\frac12)\) \(\approx\) \(1.595726622 - 1.314652573i\)
\(L(1)\) \(\approx\) \(0.9899526325 - 0.2631357818i\)
\(L(1)\) \(\approx\) \(0.9899526325 - 0.2631357818i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good3 \( 1 + (-0.743 - 0.669i)T \)
7 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (0.104 - 0.994i)T \)
17 \( 1 + (0.743 - 0.669i)T \)
19 \( 1 + (0.669 + 0.743i)T \)
23 \( 1 + (-0.994 - 0.104i)T \)
29 \( 1 + (0.669 - 0.743i)T \)
31 \( 1 + (0.309 - 0.951i)T \)
37 \( 1 + (-0.406 + 0.913i)T \)
41 \( 1 + (0.913 + 0.406i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 + (0.951 - 0.309i)T \)
53 \( 1 + (0.951 - 0.309i)T \)
59 \( 1 + (-0.104 - 0.994i)T \)
61 \( 1 + (-0.913 + 0.406i)T \)
67 \( 1 + (-0.207 - 0.978i)T \)
71 \( 1 + (-0.978 - 0.207i)T \)
73 \( 1 + (0.587 - 0.809i)T \)
79 \( 1 + (-0.309 - 0.951i)T \)
83 \( 1 + (-0.951 - 0.309i)T \)
89 \( 1 + (0.104 - 0.994i)T \)
97 \( 1 + (-0.207 + 0.978i)T \)
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

−19.55861628467054490845374480603, −18.297185009225647685225069681409, −17.7310701132983545314789494557, −17.38595013268950022834145999773, −16.54750755527182625469261114778, −15.80545989934949099765404074212, −15.21420000448645972333154870685, −14.34312985827046164703480049681, −13.92947795477320756230878454566, −12.5448477280416795913805914758, −12.208599135061225545093099072746, −11.37999174550438614459173476413, −10.56119678999925405466680645733, −10.209751421600067640882480422433, −9.29232221108830375686830734315, −8.51650246802339389212589767509, −7.41970486705794415561229015324, −6.996503107483493416484959993018, −5.801284090218342750783474598060, −5.27263367491169150377766310784, −4.34063835599496966465965682213, −3.97457742777335533966659014665, −2.757464196646252741040043210673, −1.54426162850795145682108460126, −0.81280771872280495547101981309, 0.51020775005216350058049845239, 1.2220808241613277094602860809, 2.16155036588586822593519454700, 3.03845707913671222965385684167, 4.24274991751262610606553119208, 5.07026195024658614456334383632, 5.90449594163003188477485585528, 6.16845189894527803110758364726, 7.53523451684631155310150870179, 7.865340550867980544211998312090, 8.649748176137120532985787643400, 9.709487774967981673161933140301, 10.51810824219759638140982866914, 11.34129577488388388856350895600, 11.90760022516664167077700768972, 12.256432064161989605899886526782, 13.466913348952691075087065234818, 13.9206688961312479699078159380, 14.598791935985106098394652632351, 15.70010715063021627351661031455, 16.285491266002561548209000941280, 16.98118779886154712293367861244, 17.6976644748713145154329888827, 18.431751540976116614213517744303, 18.75048268854548005050807937942

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