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Properties

Label 1-837-837.247-r1-0-0
Degree $1$
Conductor $837$
Sign $0.549 - 0.835i$
Analytic cond. $89.9481$
Root an. cond. $89.9481$
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  + (0.766 + 0.642i)2-s + (0.173 + 0.984i)4-s + (−0.939 − 0.342i)5-s + (0.173 − 0.984i)7-s + (−0.5 + 0.866i)8-s + (−0.5 − 0.866i)10-s + (0.939 − 0.342i)11-s + (−0.766 + 0.642i)13-s + (0.766 − 0.642i)14-s + (−0.939 + 0.342i)16-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.173 − 0.984i)20-s + (0.939 + 0.342i)22-s + (−0.173 − 0.984i)23-s + ⋯
L(s)  = 1  + (0.766 + 0.642i)2-s + (0.173 + 0.984i)4-s + (−0.939 − 0.342i)5-s + (0.173 − 0.984i)7-s + (−0.5 + 0.866i)8-s + (−0.5 − 0.866i)10-s + (0.939 − 0.342i)11-s + (−0.766 + 0.642i)13-s + (0.766 − 0.642i)14-s + (−0.939 + 0.342i)16-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.173 − 0.984i)20-s + (0.939 + 0.342i)22-s + (−0.173 − 0.984i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.549 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.549 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.549 - 0.835i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (247, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ 0.549 - 0.835i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.351963436 - 0.7289722025i\)
\(L(\frac12)\) \(\approx\) \(1.351963436 - 0.7289722025i\)
\(L(1)\) \(\approx\) \(1.214493696 + 0.2760035048i\)
\(L(1)\) \(\approx\) \(1.214493696 + 0.2760035048i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.766 + 0.642i)T \)
5 \( 1 + (-0.939 - 0.342i)T \)
7 \( 1 + (0.173 - 0.984i)T \)
11 \( 1 + (0.939 - 0.342i)T \)
13 \( 1 + (-0.766 + 0.642i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.173 - 0.984i)T \)
29 \( 1 + (-0.766 - 0.642i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (0.766 - 0.642i)T \)
43 \( 1 + (0.939 - 0.342i)T \)
47 \( 1 + (0.173 - 0.984i)T \)
53 \( 1 - T \)
59 \( 1 + (-0.939 - 0.342i)T \)
61 \( 1 + (-0.173 + 0.984i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (-0.766 - 0.642i)T \)
83 \( 1 + (-0.766 - 0.642i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (-0.939 + 0.342i)T \)
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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

−22.09924401327800088536614783430, −21.487641509589918267556634714709, −20.37254579132249021141892409330, −19.7270958590512938739348481929, −19.15636148037300542095001015571, −18.32954097190956408550855005394, −17.43800955190779678314066047412, −16.04190609806595465779003102758, −15.43465526448263749463391601013, −14.68273522219775060914885502490, −14.21348046931860888420512657889, −12.73655271566430722280294284228, −12.38395859704503329456702379339, −11.4093926123628715361077838096, −11.087701170057363397806479735605, −9.67543936984201928232827130360, −9.145859123305756466606983699644, −7.76758023145390245939824890518, −6.95008039231183599322586283637, −5.863779807603139125724780306943, −4.97173187145982390331448176881, −4.13330962763722958256050942055, −3.08926994424002036036137717514, −2.40481741100681113651565248838, −1.057113159922647307151645182599, 0.27968700780590260008179839180, 1.75455360101238579053573355547, 3.3216313559893643834222412475, 4.16532327239878088773050876734, 4.477449508885349210226946320464, 5.86682595342920776630144947130, 6.73041933672635461075948555448, 7.59216876359440563831521781757, 8.19693164692877869128159560277, 9.16939280018850282178635244999, 10.50422102293886799222268986769, 11.45304763192578761314851746212, 12.19054714808926066199205576395, 12.82198528075240160268467381344, 13.96382012078457725670845632913, 14.53880899742665695367192050871, 15.18804362089200821428043142662, 16.39200322603285232751010974415, 16.79191502717867924840683547820, 17.28057767526978811000612260939, 18.78891513546997197710468055974, 19.52408682992213865722409742998, 20.37800929141410898135805200886, 21.03352991917084862184142126718, 22.02863137275530190926020317087

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