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Properties

Label 1-2900-2900.1839-r0-0-0
Degree $1$
Conductor $2900$
Sign $0.951 - 0.307i$
Analytic cond. $13.4675$
Root an. cond. $13.4675$
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  + (0.587 − 0.809i)3-s + 7-s + (−0.309 − 0.951i)9-s + (−0.951 − 0.309i)11-s + (0.309 + 0.951i)13-s + (0.587 + 0.809i)17-s + (0.587 + 0.809i)19-s + (0.587 − 0.809i)21-s + (0.309 − 0.951i)23-s + (−0.951 − 0.309i)27-s + (0.587 + 0.809i)31-s + (−0.809 + 0.587i)33-s + (−0.951 + 0.309i)37-s + (0.951 + 0.309i)39-s + (0.951 − 0.309i)41-s + ⋯
L(s)  = 1  + (0.587 − 0.809i)3-s + 7-s + (−0.309 − 0.951i)9-s + (−0.951 − 0.309i)11-s + (0.309 + 0.951i)13-s + (0.587 + 0.809i)17-s + (0.587 + 0.809i)19-s + (0.587 − 0.809i)21-s + (0.309 − 0.951i)23-s + (−0.951 − 0.309i)27-s + (0.587 + 0.809i)31-s + (−0.809 + 0.587i)33-s + (−0.951 + 0.309i)37-s + (0.951 + 0.309i)39-s + (0.951 − 0.309i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.951 - 0.307i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.951 - 0.307i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2900\)    =    \(2^{2} \cdot 5^{2} \cdot 29\)
Sign: $0.951 - 0.307i$
Analytic conductor: \(13.4675\)
Root analytic conductor: \(13.4675\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2900} (1839, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2900,\ (0:\ ),\ 0.951 - 0.307i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.365486534 - 0.3725255840i\)
\(L(\frac12)\) \(\approx\) \(2.365486534 - 0.3725255840i\)
\(L(1)\) \(\approx\) \(1.421837225 - 0.2638672427i\)
\(L(1)\) \(\approx\) \(1.421837225 - 0.2638672427i\)

Euler product

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

−19.23822031357748564796037360538, −18.418424628172859081919310652531, −17.742542367996946924386854615187, −17.16646606105902587038850655771, −16.08521850971467091068363532778, −15.617438417668215730394779521385, −15.057132792076117051851993335455, −14.34295635035850150985557202330, −13.54724914797579585069489501420, −13.1158254433741061031318087182, −11.858430192596811551009091731383, −11.29035017415589237190967634252, −10.48406423722932076229863974170, −9.96301778575251158669498594782, −9.111281124827281101518511143684, −8.360515806087658821362214434078, −7.694075479042304110387233603590, −7.19793984752932167165166893170, −5.47275561226280742730667422902, −5.34413056606069470026123863980, −4.5007476925569369618248016337, −3.53019520412353992025080091277, −2.789893426789187928824335420549, −2.05435992777697669445173463379, −0.789022349142371370391109678359, 1.02650050723591511955478686353, 1.68088101600834978135040876681, 2.5275478552027971680038820955, 3.38456600434671684909762632266, 4.28773769164968481805470116781, 5.23032146821000896053774907097, 6.045617373895940491105137387715, 6.84060244030725128605948091101, 7.71755927974357588236806771560, 8.22920776818593256161922915307, 8.73899935664609802991654411667, 9.74486673663861487779680091496, 10.63501597714133403770200434373, 11.33076786357081739347631215103, 12.183407087055153837903151528098, 12.67026872733211711568223503664, 13.595850250717179863991972180865, 14.24043404878370434731771295481, 14.56387736729293957313851897954, 15.520395519755300905835167312614, 16.308541550419228377198295129666, 17.147803148959402567421975764484, 17.8695549287893997331458980312, 18.53756679908736473314181553823, 18.9252128506659699921027882816

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