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Properties

Label 1-1400-1400.1229-r0-0-0
Degree $1$
Conductor $1400$
Sign $-0.876 - 0.481i$
Analytic cond. $6.50157$
Root an. cond. $6.50157$
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  + (0.669 − 0.743i)3-s + (−0.104 − 0.994i)9-s + (0.104 − 0.994i)11-s + (−0.809 + 0.587i)13-s + (0.978 − 0.207i)17-s + (−0.669 − 0.743i)19-s + (−0.913 + 0.406i)23-s + (−0.809 − 0.587i)27-s + (−0.309 − 0.951i)29-s + (−0.978 + 0.207i)31-s + (−0.669 − 0.743i)33-s + (−0.104 − 0.994i)37-s + (−0.104 + 0.994i)39-s + (−0.809 + 0.587i)41-s + 43-s + ⋯
L(s)  = 1  + (0.669 − 0.743i)3-s + (−0.104 − 0.994i)9-s + (0.104 − 0.994i)11-s + (−0.809 + 0.587i)13-s + (0.978 − 0.207i)17-s + (−0.669 − 0.743i)19-s + (−0.913 + 0.406i)23-s + (−0.809 − 0.587i)27-s + (−0.309 − 0.951i)29-s + (−0.978 + 0.207i)31-s + (−0.669 − 0.743i)33-s + (−0.104 − 0.994i)37-s + (−0.104 + 0.994i)39-s + (−0.809 + 0.587i)41-s + 43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.876 - 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.876 - 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.876 - 0.481i$
Analytic conductor: \(6.50157\)
Root analytic conductor: \(6.50157\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (1229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (0:\ ),\ -0.876 - 0.481i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3167930328 - 1.235220452i\)
\(L(\frac12)\) \(\approx\) \(0.3167930328 - 1.235220452i\)
\(L(1)\) \(\approx\) \(1.001687194 - 0.5058297582i\)
\(L(1)\) \(\approx\) \(1.001687194 - 0.5058297582i\)

Euler product

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

−20.994875319453920834622688409237, −20.28971230358097688623056030223, −19.89524258908990563731047934153, −18.91723241523659313027280042312, −18.217060479540391534476228889075, −17.04037826178008692622734643046, −16.69979214017096027110114259212, −15.63622454586001482472916052229, −14.939379904275262902515411575210, −14.503955590514428495467046794769, −13.67114334315865732205646009406, −12.50405804589063497217848318093, −12.18926478559514733828406620913, −10.7279342829731631714959083411, −10.24824696424666456360613064967, −9.572860788047886504113631124851, −8.717358960880700733078231098210, −7.807705362871167858970207900670, −7.2715769402553911380030730714, −5.91955532854537313272687388401, −5.09261330054374224735617823893, −4.24449668837972931782311306142, −3.47183522820041810405025861849, −2.47201755660662670432757485908, −1.63288881431153116625966818745, 0.410696712330584271169153141457, 1.670584202213856435344679155748, 2.51170118996737373356391943257, 3.416645046898306969080542503888, 4.28143003326153927293912110229, 5.58352045404106681401397962313, 6.28598486113612180007125631865, 7.31042800446796913030814378161, 7.80675758311818206616424917487, 8.834451458186838400303247275307, 9.3487441885739666826258739317, 10.35979647374116675779283791121, 11.46198902247432580786529129111, 12.06216848196035138920605072565, 12.894344315340640915344509272722, 13.70301605396865811682630523585, 14.27737314249377642686396590075, 14.94334044671142173795104276747, 15.9268287275496453492487127448, 16.78753173500800700025978407027, 17.50746259491337586834083113, 18.41692403993551557028176332332, 19.07754920304233599845388914641, 19.56383066473929202345035308843, 20.32456675814888382916400888526

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