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Properties

Label 1-1148-1148.403-r0-0-0
Degree $1$
Conductor $1148$
Sign $-0.940 - 0.340i$
Analytic cond. $5.33128$
Root an. cond. $5.33128$
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  + (−0.258 + 0.965i)3-s + (0.207 − 0.978i)5-s + (−0.866 − 0.5i)9-s + (−0.838 − 0.544i)11-s + (−0.156 − 0.987i)13-s + (0.891 + 0.453i)15-s + (−0.544 + 0.838i)17-s + (0.777 + 0.629i)19-s + (0.913 − 0.406i)23-s + (−0.913 − 0.406i)25-s + (0.707 − 0.707i)27-s + (−0.453 + 0.891i)29-s + (−0.978 + 0.207i)31-s + (0.743 − 0.669i)33-s + (−0.978 − 0.207i)37-s + ⋯
L(s)  = 1  + (−0.258 + 0.965i)3-s + (0.207 − 0.978i)5-s + (−0.866 − 0.5i)9-s + (−0.838 − 0.544i)11-s + (−0.156 − 0.987i)13-s + (0.891 + 0.453i)15-s + (−0.544 + 0.838i)17-s + (0.777 + 0.629i)19-s + (0.913 − 0.406i)23-s + (−0.913 − 0.406i)25-s + (0.707 − 0.707i)27-s + (−0.453 + 0.891i)29-s + (−0.978 + 0.207i)31-s + (0.743 − 0.669i)33-s + (−0.978 − 0.207i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.940 - 0.340i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.940 - 0.340i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.940 - 0.340i$
Analytic conductor: \(5.33128\)
Root analytic conductor: \(5.33128\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (403, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1148,\ (0:\ ),\ -0.940 - 0.340i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02431631565 - 0.1385859944i\)
\(L(\frac12)\) \(\approx\) \(0.02431631565 - 0.1385859944i\)
\(L(1)\) \(\approx\) \(0.7338123589 + 0.03122408260i\)
\(L(1)\) \(\approx\) \(0.7338123589 + 0.03122408260i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good3 \( 1 + (-0.258 + 0.965i)T \)
5 \( 1 + (0.207 - 0.978i)T \)
11 \( 1 + (-0.838 - 0.544i)T \)
13 \( 1 + (-0.156 - 0.987i)T \)
17 \( 1 + (-0.544 + 0.838i)T \)
19 \( 1 + (0.777 + 0.629i)T \)
23 \( 1 + (0.913 - 0.406i)T \)
29 \( 1 + (-0.453 + 0.891i)T \)
31 \( 1 + (-0.978 + 0.207i)T \)
37 \( 1 + (-0.978 - 0.207i)T \)
43 \( 1 + (0.587 - 0.809i)T \)
47 \( 1 + (-0.358 + 0.933i)T \)
53 \( 1 + (-0.998 + 0.0523i)T \)
59 \( 1 + (0.104 - 0.994i)T \)
61 \( 1 + (-0.994 + 0.104i)T \)
67 \( 1 + (0.0523 + 0.998i)T \)
71 \( 1 + (-0.891 + 0.453i)T \)
73 \( 1 + (-0.866 + 0.5i)T \)
79 \( 1 + (-0.965 + 0.258i)T \)
83 \( 1 - T \)
89 \( 1 + (0.629 - 0.777i)T \)
97 \( 1 + (-0.891 - 0.453i)T \)
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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

−21.76258083447104992634367454437, −20.871577960922763526156563966472, −19.94071766043058662523088197000, −19.1055809532592829236891243910, −18.530984639698552000940996819520, −17.90675441372378748582261418256, −17.30086924381186181251125211285, −16.271212648583943806295736624114, −15.34755580381674124931614832296, −14.54201566456279802713255052228, −13.611040781334035516283421379704, −13.30189072829708128345471160111, −12.13617332454317478793521627832, −11.38752993056553165594068251004, −10.8643666738479631672941505249, −9.71924480743488892636462789812, −8.96078538091114266590931168690, −7.560176055142312635592491651329, −7.26953512213054490296471015368, −6.501742306790510487278346578758, −5.525282639042832600481432498975, −4.665635283268884871932982308863, −3.1785367415150129008499970359, −2.43589600282348341101709983727, −1.62222172920776487367545901062, 0.05700658482160016899972129765, 1.419851673128572114850656218986, 2.869113406252138407539247176023, 3.68602738542830734683032347545, 4.749885811326899451578131689641, 5.43423112581846908280482161681, 5.931944765508593390005408399, 7.41585686573892755717030770518, 8.458320330509640372068700620322, 8.92258104554774621259213052905, 9.92964862613301677155330571493, 10.61913066225751011727406935501, 11.28911725417783613850064127179, 12.5384895109161720655291483430, 12.88314877556496492159120380985, 14.010652446874822399604162469791, 14.85929912998471884849488336331, 15.80949682176100610878649139385, 16.12833170511048808187993501879, 17.09567057725886553282627449617, 17.5773553857988772856649697440, 18.57268553218601586636359330528, 19.66438985131475778380207550919, 20.53307997474293369416186076169, 20.7578242286044680400273997549

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