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Properties

Label 1-1840-1840.147-r0-0-0
Degree $1$
Conductor $1840$
Sign $0.976 - 0.217i$
Analytic cond. $8.54492$
Root an. cond. $8.54492$
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  + (0.142 − 0.989i)3-s + (0.755 − 0.654i)7-s + (−0.959 − 0.281i)9-s + (−0.540 + 0.841i)11-s + (−0.654 + 0.755i)13-s + (−0.909 + 0.415i)17-s + (0.909 + 0.415i)19-s + (−0.540 − 0.841i)21-s + (−0.415 + 0.909i)27-s + (0.909 − 0.415i)29-s + (0.142 + 0.989i)31-s + (0.755 + 0.654i)33-s + (−0.959 − 0.281i)37-s + (0.654 + 0.755i)39-s + (0.959 − 0.281i)41-s + ⋯
L(s)  = 1  + (0.142 − 0.989i)3-s + (0.755 − 0.654i)7-s + (−0.959 − 0.281i)9-s + (−0.540 + 0.841i)11-s + (−0.654 + 0.755i)13-s + (−0.909 + 0.415i)17-s + (0.909 + 0.415i)19-s + (−0.540 − 0.841i)21-s + (−0.415 + 0.909i)27-s + (0.909 − 0.415i)29-s + (0.142 + 0.989i)31-s + (0.755 + 0.654i)33-s + (−0.959 − 0.281i)37-s + (0.654 + 0.755i)39-s + (0.959 − 0.281i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.217i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.217i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $0.976 - 0.217i$
Analytic conductor: \(8.54492\)
Root analytic conductor: \(8.54492\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1840,\ (0:\ ),\ 0.976 - 0.217i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.519709652 - 0.1672261963i\)
\(L(\frac12)\) \(\approx\) \(1.519709652 - 0.1672261963i\)
\(L(1)\) \(\approx\) \(1.075992852 - 0.2407011863i\)
\(L(1)\) \(\approx\) \(1.075992852 - 0.2407011863i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good3 \( 1 + (0.142 - 0.989i)T \)
7 \( 1 + (0.755 - 0.654i)T \)
11 \( 1 + (-0.540 + 0.841i)T \)
13 \( 1 + (-0.654 + 0.755i)T \)
17 \( 1 + (-0.909 + 0.415i)T \)
19 \( 1 + (0.909 + 0.415i)T \)
29 \( 1 + (0.909 - 0.415i)T \)
31 \( 1 + (0.142 + 0.989i)T \)
37 \( 1 + (-0.959 - 0.281i)T \)
41 \( 1 + (0.959 - 0.281i)T \)
43 \( 1 + (-0.142 + 0.989i)T \)
47 \( 1 - iT \)
53 \( 1 + (0.654 + 0.755i)T \)
59 \( 1 + (0.755 + 0.654i)T \)
61 \( 1 + (0.989 - 0.142i)T \)
67 \( 1 + (0.841 - 0.540i)T \)
71 \( 1 + (0.841 - 0.540i)T \)
73 \( 1 + (0.909 + 0.415i)T \)
79 \( 1 + (-0.654 + 0.755i)T \)
83 \( 1 + (0.959 + 0.281i)T \)
89 \( 1 + (-0.142 + 0.989i)T \)
97 \( 1 + (-0.281 - 0.959i)T \)
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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

−20.36599380561477141280498486825, −19.55884527283061115022053057008, −18.71451464787854791895241577921, −17.75568536606188600737050792788, −17.399143246092811338083885150812, −16.23722452074856220317655254072, −15.71735470877211354753999073011, −15.172652513138775384101287332665, −14.32007458009984589895073589768, −13.725369584647950000460868040043, −12.74936110412265816233375938644, −11.66304327082242052806269302234, −11.24265443799265219817283516062, −10.418299023739435100363786780078, −9.653120191298457006519476853756, −8.7943138484797722483639624836, −8.26941781851042842482139216501, −7.41333397381528871155778177110, −6.15281520873571182716257872802, −5.153856016513757004964718304100, −5.02643648071025056799517441387, −3.81617022138318244948501252124, −2.82722733993162021646538077647, −2.31781796583111046570279493171, −0.63136207339806593331327403752, 0.94593389265909059112750093312, 1.902747433060953871239152829230, 2.50100060494925293173499006637, 3.776675881460956487367137612756, 4.71178655593072250438506560028, 5.44184250254219920907521593831, 6.70528139784568258607073482127, 7.10220534796569469405132977173, 7.89801606224984782909790456664, 8.55354121225275971586837084474, 9.5589060165599509905658433777, 10.43855937664087948464226246168, 11.27684597838615282541928431630, 12.051692549686105691398225345771, 12.612732799962133911901123862988, 13.586125984237913452047502752217, 14.07333205638151436257629231395, 14.74958327426953586653813407996, 15.63602229014734676913307340203, 16.63908648268259204913520648243, 17.47562695980785941318146217733, 17.84496607298240203670774634287, 18.52088214745671485582758950018, 19.64749330853678860443715112413, 19.81872743084873231802271949413

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