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Properties

Label 1-437-437.73-r0-0-0
Degree $1$
Conductor $437$
Sign $-0.694 + 0.719i$
Analytic cond. $2.02941$
Root an. cond. $2.02941$
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  + (−0.857 − 0.513i)2-s + (0.296 − 0.954i)3-s + (0.472 + 0.881i)4-s + (−0.0792 − 0.996i)5-s + (−0.745 + 0.666i)6-s + (0.235 − 0.971i)7-s + (0.0475 − 0.998i)8-s + (−0.823 − 0.567i)9-s + (−0.444 + 0.895i)10-s + (−0.327 + 0.945i)11-s + (0.981 − 0.189i)12-s + (−0.553 − 0.832i)13-s + (−0.701 + 0.712i)14-s + (−0.975 − 0.220i)15-s + (−0.553 + 0.832i)16-s + (−0.999 + 0.0317i)17-s + ⋯
L(s)  = 1  + (−0.857 − 0.513i)2-s + (0.296 − 0.954i)3-s + (0.472 + 0.881i)4-s + (−0.0792 − 0.996i)5-s + (−0.745 + 0.666i)6-s + (0.235 − 0.971i)7-s + (0.0475 − 0.998i)8-s + (−0.823 − 0.567i)9-s + (−0.444 + 0.895i)10-s + (−0.327 + 0.945i)11-s + (0.981 − 0.189i)12-s + (−0.553 − 0.832i)13-s + (−0.701 + 0.712i)14-s + (−0.975 − 0.220i)15-s + (−0.553 + 0.832i)16-s + (−0.999 + 0.0317i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 437 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.694 + 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 437 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.694 + 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(437\)    =    \(19 \cdot 23\)
Sign: $-0.694 + 0.719i$
Analytic conductor: \(2.02941\)
Root analytic conductor: \(2.02941\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{437} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 437,\ (0:\ ),\ -0.694 + 0.719i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2163936242 - 0.5095176601i\)
\(L(\frac12)\) \(\approx\) \(-0.2163936242 - 0.5095176601i\)
\(L(1)\) \(\approx\) \(0.4076227077 - 0.5018153631i\)
\(L(1)\) \(\approx\) \(0.4076227077 - 0.5018153631i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.857 - 0.513i)T \)
3 \( 1 + (0.296 - 0.954i)T \)
5 \( 1 + (-0.0792 - 0.996i)T \)
7 \( 1 + (0.235 - 0.971i)T \)
11 \( 1 + (-0.327 + 0.945i)T \)
13 \( 1 + (-0.553 - 0.832i)T \)
17 \( 1 + (-0.999 + 0.0317i)T \)
29 \( 1 + (0.950 + 0.312i)T \)
31 \( 1 + (-0.888 + 0.458i)T \)
37 \( 1 + (0.415 - 0.909i)T \)
41 \( 1 + (0.902 + 0.429i)T \)
43 \( 1 + (-0.975 + 0.220i)T \)
47 \( 1 + (0.173 - 0.984i)T \)
53 \( 1 + (0.805 - 0.592i)T \)
59 \( 1 + (0.110 - 0.993i)T \)
61 \( 1 + (-0.975 - 0.220i)T \)
67 \( 1 + (0.630 + 0.776i)T \)
71 \( 1 + (0.873 - 0.486i)T \)
73 \( 1 + (-0.745 - 0.666i)T \)
79 \( 1 + (-0.916 + 0.400i)T \)
83 \( 1 + (0.580 + 0.814i)T \)
89 \( 1 + (0.991 + 0.126i)T \)
97 \( 1 + (-0.823 + 0.567i)T \)
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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

−24.78889261893766139443702366264, −24.02696171582356914274387625931, −22.85336497070457838313409647669, −21.83524503379468732380523394996, −21.42157540872782178035668975612, −20.13588944908808426058110450438, −19.24295678988571246409419439740, −18.64962883599805106442376987194, −17.7469355505272406804895450993, −16.71353800539411109909381679477, −15.82250279892456569044432931800, −15.235860371381618702302970592416, −14.494014195736825130512218791131, −13.72929038368890193491481197942, −11.69969087003798976737426478569, −11.13210449798178355430514052831, −10.281511392393408043585894707841, −9.301934343351669667685549022335, −8.638838433352906424564722025365, −7.67881223715291898428557834586, −6.44484968653029365180487923801, −5.63735541082550937251074437369, −4.4662137324722759757716631153, −2.89770858675026122480331658390, −2.19490421808652169120800015541, 0.387845679954732394289763941478, 1.499562696863072794074396070198, 2.44876241934964320551789118502, 3.83480279316960470425360363971, 5.02210661566801751135000378876, 6.6946812310596234123833336886, 7.52152911435179198907089992551, 8.156468594898511895226960162808, 9.0938898767985243294114925575, 10.0719014046193610429896268930, 11.10599735508830395748565832271, 12.19337721621090489981948383888, 12.83143634627304491374102012326, 13.438614348952013270023128003115, 14.80468321193613238516226857510, 15.983209593394933798095968491644, 17.00067637645335268958983690061, 17.68426953612414634853444226590, 18.13400254105705041664865736954, 19.655948609089346004989457906632, 19.9417598649687982575891854759, 20.452199313938702175980981539184, 21.49677885426536075971137323408, 22.89108304163248390692675241423, 23.68023370531197690323365404743

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