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Properties

Label 1-5600-5600.3979-r0-0-0
Degree $1$
Conductor $5600$
Sign $-0.209 - 0.977i$
Analytic cond. $26.0062$
Root an. cond. $26.0062$
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  + (0.0523 + 0.998i)3-s + (−0.994 + 0.104i)9-s + (0.777 − 0.629i)11-s + (0.156 − 0.987i)13-s + (0.978 − 0.207i)17-s + (−0.998 − 0.0523i)19-s + (−0.406 − 0.913i)23-s + (−0.156 − 0.987i)27-s + (0.891 + 0.453i)29-s + (−0.978 + 0.207i)31-s + (0.669 + 0.743i)33-s + (0.629 − 0.777i)37-s + (0.994 + 0.104i)39-s + (−0.587 − 0.809i)41-s + (0.707 + 0.707i)43-s + ⋯
L(s)  = 1  + (0.0523 + 0.998i)3-s + (−0.994 + 0.104i)9-s + (0.777 − 0.629i)11-s + (0.156 − 0.987i)13-s + (0.978 − 0.207i)17-s + (−0.998 − 0.0523i)19-s + (−0.406 − 0.913i)23-s + (−0.156 − 0.987i)27-s + (0.891 + 0.453i)29-s + (−0.978 + 0.207i)31-s + (0.669 + 0.743i)33-s + (0.629 − 0.777i)37-s + (0.994 + 0.104i)39-s + (−0.587 − 0.809i)41-s + (0.707 + 0.707i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.209 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.209 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5600\)    =    \(2^{5} \cdot 5^{2} \cdot 7\)
Sign: $-0.209 - 0.977i$
Analytic conductor: \(26.0062\)
Root analytic conductor: \(26.0062\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5600} (3979, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5600,\ (0:\ ),\ -0.209 - 0.977i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5001511794 - 0.6186030974i\)
\(L(\frac12)\) \(\approx\) \(0.5001511794 - 0.6186030974i\)
\(L(1)\) \(\approx\) \(0.9463059118 + 0.1041982691i\)
\(L(1)\) \(\approx\) \(0.9463059118 + 0.1041982691i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.0523 + 0.998i)T \)
11 \( 1 + (0.777 - 0.629i)T \)
13 \( 1 + (0.156 - 0.987i)T \)
17 \( 1 + (0.978 - 0.207i)T \)
19 \( 1 + (-0.998 - 0.0523i)T \)
23 \( 1 + (-0.406 - 0.913i)T \)
29 \( 1 + (0.891 + 0.453i)T \)
31 \( 1 + (-0.978 + 0.207i)T \)
37 \( 1 + (0.629 - 0.777i)T \)
41 \( 1 + (-0.587 - 0.809i)T \)
43 \( 1 + (0.707 + 0.707i)T \)
47 \( 1 + (-0.978 - 0.207i)T \)
53 \( 1 + (-0.998 + 0.0523i)T \)
59 \( 1 + (-0.358 - 0.933i)T \)
61 \( 1 + (-0.358 + 0.933i)T \)
67 \( 1 + (-0.544 + 0.838i)T \)
71 \( 1 + (0.951 - 0.309i)T \)
73 \( 1 + (0.994 + 0.104i)T \)
79 \( 1 + (-0.978 - 0.207i)T \)
83 \( 1 + (-0.453 - 0.891i)T \)
89 \( 1 + (-0.406 - 0.913i)T \)
97 \( 1 + (0.309 + 0.951i)T \)
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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

−18.17762818969710797782805452327, −17.17873273253483441162758857449, −17.03675366383205124920011021774, −16.218268653968517368939358421703, −15.191005859009903371387046504166, −14.63431542707313176021807863320, −14.01396633713107001801606123161, −13.48047188689274493903536300617, −12.56380736383318812261986740314, −12.21098141238854617088795870110, −11.50839031189930046871207514792, −10.928146380133761289508644962469, −9.79034663195395940815823960990, −9.3735415454616451412208732972, −8.44589116861008294597792025890, −7.9109212039936533475808123266, −7.14690688883386325274272306194, −6.47195566738116206173674966807, −6.050122781612768089360411552947, −5.05773127113387692039082051819, −4.18350457188280588262244812465, −3.47209189256994469267832833994, −2.507635995799215140914267914080, −1.66806577772393125365557214457, −1.27326818040479806335525098560, 0.20102577721184264216195075482, 1.2576916893656131366328836167, 2.45407156298774007883692352667, 3.1988894248665099598978910505, 3.77928727540408190422317461894, 4.52946458191191356118202258850, 5.31268527275281131034407100409, 5.96853882705066175346294831541, 6.58441333737268932847716781493, 7.72379013841212797121812864221, 8.366915169942568594749774932829, 8.92978636483199518410825693853, 9.6424153817442419155061904237, 10.42898528324672478563820909066, 10.79335024506109266220767807744, 11.55325541825529333808713038950, 12.34998865088021012121229771624, 12.9400747027620544573924115943, 13.97766779218192257454962958014, 14.514485214188606258308393914293, 14.85833462196250696381407361134, 15.90487257410728381254571820059, 16.18949596250250801458906102624, 16.94351534853769558203519560253, 17.45228322483227059398100286079

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