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Properties

Label 1-799-799.563-r1-0-0
Degree $1$
Conductor $799$
Sign $-0.739 - 0.673i$
Analytic cond. $85.8644$
Root an. cond. $85.8644$
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  + i·2-s + (0.707 − 0.707i)3-s − 4-s + (0.707 − 0.707i)5-s + (0.707 + 0.707i)6-s + (−0.707 − 0.707i)7-s − i·8-s − i·9-s + (0.707 + 0.707i)10-s + (−0.707 − 0.707i)11-s + (−0.707 + 0.707i)12-s + 13-s + (0.707 − 0.707i)14-s − i·15-s + 16-s + ⋯
L(s)  = 1  + i·2-s + (0.707 − 0.707i)3-s − 4-s + (0.707 − 0.707i)5-s + (0.707 + 0.707i)6-s + (−0.707 − 0.707i)7-s − i·8-s − i·9-s + (0.707 + 0.707i)10-s + (−0.707 − 0.707i)11-s + (−0.707 + 0.707i)12-s + 13-s + (0.707 − 0.707i)14-s − i·15-s + 16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.739 - 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.739 - 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(799\)    =    \(17 \cdot 47\)
Sign: $-0.739 - 0.673i$
Analytic conductor: \(85.8644\)
Root analytic conductor: \(85.8644\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{799} (563, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 799,\ (1:\ ),\ -0.739 - 0.673i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6559479688 - 1.694027628i\)
\(L(\frac12)\) \(\approx\) \(0.6559479688 - 1.694027628i\)
\(L(1)\) \(\approx\) \(1.153750044 - 0.2762470639i\)
\(L(1)\) \(\approx\) \(1.153750044 - 0.2762470639i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
47 \( 1 \)
good2 \( 1 + iT \)
3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (0.707 - 0.707i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
11 \( 1 + (-0.707 - 0.707i)T \)
13 \( 1 + T \)
19 \( 1 - iT \)
23 \( 1 + (-0.707 - 0.707i)T \)
29 \( 1 + (0.707 - 0.707i)T \)
31 \( 1 + (-0.707 + 0.707i)T \)
37 \( 1 + (0.707 - 0.707i)T \)
41 \( 1 + (0.707 + 0.707i)T \)
43 \( 1 + iT \)
53 \( 1 + iT \)
59 \( 1 - iT \)
61 \( 1 + (-0.707 - 0.707i)T \)
67 \( 1 - T \)
71 \( 1 + (0.707 - 0.707i)T \)
73 \( 1 + (0.707 - 0.707i)T \)
79 \( 1 + (0.707 + 0.707i)T \)
83 \( 1 + iT \)
89 \( 1 - T \)
97 \( 1 + (-0.707 + 0.707i)T \)
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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

−22.2553788379414263285389351775, −21.37940681767440124586218909857, −20.92048246553630383789272512493, −20.10928780287896525147673179794, −19.23420310773287442658480268716, −18.43261566647854485367557527979, −18.04393318689371038874435745814, −16.73769819855562493415038859055, −15.72815709428687139556679666479, −14.95922727918312381721196015928, −14.11559643838825519038513292619, −13.390152828060070007177251873987, −12.69707194858081298982272196845, −11.5673161863170954399550367461, −10.5123165139224272829229895273, −10.08672232999981282009205490718, −9.3563794545279069520322413302, −8.5793413240842340445543094634, −7.53877115478545827089281744109, −5.99810706843838044991331244335, −5.31365724964909131405468454699, −4.00349132761097698069791160426, −3.258324524689030025352357071, −2.445396317118864999391792194171, −1.70329348265952270168151479576, 0.3859543713626942983230838495, 1.11699753922068237487791735267, 2.67009085430241878057184066886, 3.74027835908107547679711852704, 4.770589811045002616062093352, 6.12859689619088811969243234542, 6.34714540334655637025194166783, 7.58388193623329459161357618870, 8.293092387250385105713294354754, 9.0590842816964007390897396288, 9.74752298787372731471878759181, 10.86091383108016594343599104979, 12.582819916309918719656369958429, 13.002390365233039411257396409647, 13.81503879127203993554343349027, 14.03780165771503367646135225964, 15.42231787760101456073263797688, 16.159766684020433664322090142098, 16.77219485022648190989202594002, 17.933825277922672681535699834817, 18.19633672318878788407658388627, 19.34025178282260210321864130045, 20.00697430561870047558879830878, 21.01705629354632144859742117565, 21.71225149472620342468431289721

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