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Properties

Label 1-497-497.136-r0-0-0
Degree $1$
Conductor $497$
Sign $0.355 + 0.934i$
Analytic cond. $2.30805$
Root an. cond. $2.30805$
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  + (0.887 + 0.460i)2-s + (0.946 + 0.323i)3-s + (0.575 + 0.817i)4-s + (−0.669 − 0.743i)5-s + (0.691 + 0.722i)6-s + (0.134 + 0.990i)8-s + (0.791 + 0.611i)9-s + (−0.251 − 0.967i)10-s + (−0.525 + 0.850i)11-s + (0.280 + 0.959i)12-s + (0.473 − 0.880i)13-s + (−0.393 − 0.919i)15-s + (−0.337 + 0.941i)16-s + (0.913 + 0.406i)17-s + (0.420 + 0.907i)18-s + (0.599 + 0.800i)19-s + ⋯
L(s)  = 1  + (0.887 + 0.460i)2-s + (0.946 + 0.323i)3-s + (0.575 + 0.817i)4-s + (−0.669 − 0.743i)5-s + (0.691 + 0.722i)6-s + (0.134 + 0.990i)8-s + (0.791 + 0.611i)9-s + (−0.251 − 0.967i)10-s + (−0.525 + 0.850i)11-s + (0.280 + 0.959i)12-s + (0.473 − 0.880i)13-s + (−0.393 − 0.919i)15-s + (−0.337 + 0.941i)16-s + (0.913 + 0.406i)17-s + (0.420 + 0.907i)18-s + (0.599 + 0.800i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 497 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.355 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 497 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.355 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(497\)    =    \(7 \cdot 71\)
Sign: $0.355 + 0.934i$
Analytic conductor: \(2.30805\)
Root analytic conductor: \(2.30805\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{497} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 497,\ (0:\ ),\ 0.355 + 0.934i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.397702162 + 1.654081563i\)
\(L(\frac12)\) \(\approx\) \(2.397702162 + 1.654081563i\)
\(L(1)\) \(\approx\) \(1.967377019 + 0.8234844180i\)
\(L(1)\) \(\approx\) \(1.967377019 + 0.8234844180i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
71 \( 1 \)
good2 \( 1 + (0.887 + 0.460i)T \)
3 \( 1 + (0.946 + 0.323i)T \)
5 \( 1 + (-0.669 - 0.743i)T \)
11 \( 1 + (-0.525 + 0.850i)T \)
13 \( 1 + (0.473 - 0.880i)T \)
17 \( 1 + (0.913 + 0.406i)T \)
19 \( 1 + (0.599 + 0.800i)T \)
23 \( 1 + (-0.365 - 0.930i)T \)
29 \( 1 + (0.858 + 0.512i)T \)
31 \( 1 + (-0.337 - 0.941i)T \)
37 \( 1 + (-0.988 + 0.149i)T \)
41 \( 1 + (-0.900 + 0.433i)T \)
43 \( 1 + (0.936 - 0.351i)T \)
47 \( 1 + (-0.946 + 0.323i)T \)
53 \( 1 + (-0.420 - 0.907i)T \)
59 \( 1 + (0.971 + 0.237i)T \)
61 \( 1 + (0.712 - 0.701i)T \)
67 \( 1 + (-0.575 - 0.817i)T \)
73 \( 1 + (0.842 - 0.538i)T \)
79 \( 1 + (-0.925 - 0.379i)T \)
83 \( 1 + (0.691 - 0.722i)T \)
89 \( 1 + (-0.575 + 0.817i)T \)
97 \( 1 + (-0.900 - 0.433i)T \)
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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

−23.75465351228717420649006863837, −22.71119777096028480523416772532, −21.59986688198913987155623794083, −21.14276732212107748652478526577, −20.11523371390318274217157348197, −19.27568230082918734769086704001, −18.89796777172980458747433369749, −17.99561437315794169077122821791, −15.97504704549030184746189138082, −15.78715019474128078692749482785, −14.60080167301738686342831174822, −13.948509407863142402242681671750, −13.44158138476355458574765311634, −12.16921773319076785098557519318, −11.53119425090486668365635140371, −10.55046295703004507893741186677, −9.5602955741311464695776682712, −8.371059812646716446071209501837, −7.31181108174642965245229872717, −6.62573532551369803023218395436, −5.35437818200981045615443369391, −4.01021161422361822546564924822, −3.28890256494457021694548523010, −2.57865111710016815795801903521, −1.21675872801497985342108619365, 1.70299292971066502471238513877, 3.05955161245354140629714770664, 3.808643126602795817916629945, 4.74736011411328525302215837974, 5.55718750934821213711686664593, 7.057112446781339929395100395887, 8.13729725788238910925511905088, 8.21550755221834020743527350893, 9.74623577800445459829784928017, 10.72842432216315942528202688908, 12.169047368304678284659942446120, 12.659243223566685565147258163174, 13.50741700119877836977159410298, 14.54535175820144871160123563981, 15.168462792748926364906816188046, 15.98639145915808152525577844255, 16.50018832829036669867050229608, 17.76868935279739155254271688164, 18.94995873108399377998947700299, 20.068102336341367236185759464239, 20.6192882370490760311447472664, 21.017541907879597609855343039617, 22.30647224611578660830247964267, 23.02687280878453348533368345356, 23.899896705469702665225336424

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