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Properties

Label 1-163-163.155-r0-0-0
Degree $1$
Conductor $163$
Sign $-0.296 - 0.955i$
Analytic cond. $0.756968$
Root an. cond. $0.756968$
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  + (−0.993 − 0.116i)2-s + (−0.686 + 0.727i)3-s + (0.973 + 0.230i)4-s + (0.173 − 0.984i)5-s + (0.766 − 0.642i)6-s + (0.597 − 0.802i)7-s + (−0.939 − 0.342i)8-s + (−0.0581 − 0.998i)9-s + (−0.286 + 0.957i)10-s + (−0.686 + 0.727i)11-s + (−0.835 + 0.549i)12-s + (−0.939 + 0.342i)13-s + (−0.686 + 0.727i)14-s + (0.597 + 0.802i)15-s + (0.893 + 0.448i)16-s + (−0.939 − 0.342i)17-s + ⋯
L(s)  = 1  + (−0.993 − 0.116i)2-s + (−0.686 + 0.727i)3-s + (0.973 + 0.230i)4-s + (0.173 − 0.984i)5-s + (0.766 − 0.642i)6-s + (0.597 − 0.802i)7-s + (−0.939 − 0.342i)8-s + (−0.0581 − 0.998i)9-s + (−0.286 + 0.957i)10-s + (−0.686 + 0.727i)11-s + (−0.835 + 0.549i)12-s + (−0.939 + 0.342i)13-s + (−0.686 + 0.727i)14-s + (0.597 + 0.802i)15-s + (0.893 + 0.448i)16-s + (−0.939 − 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.296 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.296 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(163\)
Sign: $-0.296 - 0.955i$
Analytic conductor: \(0.756968\)
Root analytic conductor: \(0.756968\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{163} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 163,\ (0:\ ),\ -0.296 - 0.955i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2273057187 - 0.3085637550i\)
\(L(\frac12)\) \(\approx\) \(0.2273057187 - 0.3085637550i\)
\(L(1)\) \(\approx\) \(0.4854872937 - 0.1183068561i\)
\(L(1)\) \(\approx\) \(0.4854872937 - 0.1183068561i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad163 \( 1 \)
good2 \( 1 + (-0.993 - 0.116i)T \)
3 \( 1 + (-0.686 + 0.727i)T \)
5 \( 1 + (0.173 - 0.984i)T \)
7 \( 1 + (0.597 - 0.802i)T \)
11 \( 1 + (-0.686 + 0.727i)T \)
13 \( 1 + (-0.939 + 0.342i)T \)
17 \( 1 + (-0.939 - 0.342i)T \)
19 \( 1 + (0.396 - 0.918i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.993 + 0.116i)T \)
31 \( 1 + (0.173 - 0.984i)T \)
37 \( 1 + (0.766 + 0.642i)T \)
41 \( 1 + (0.973 - 0.230i)T \)
43 \( 1 + (-0.286 - 0.957i)T \)
47 \( 1 + (-0.993 - 0.116i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + T \)
61 \( 1 + (-0.939 + 0.342i)T \)
67 \( 1 + (0.973 - 0.230i)T \)
71 \( 1 + (-0.286 - 0.957i)T \)
73 \( 1 + (0.973 + 0.230i)T \)
79 \( 1 + (-0.0581 + 0.998i)T \)
83 \( 1 + (0.597 - 0.802i)T \)
89 \( 1 + (-0.686 - 0.727i)T \)
97 \( 1 + (-0.0581 + 0.998i)T \)
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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

−28.01761113195954584340891466291, −27.077036808468447327330928847435, −26.23366181181060873641402984775, −24.9747960102419284519933508997, −24.47229976107088539337779912764, −23.395037847896787371387390737607, −22.09876315862965587601262912490, −21.33882956208119849003661341010, −19.701963982675657012640052661265, −18.8850213343681990510471842447, −18.04162983781551989453944812150, −17.64911335310765972106900774153, −16.322281314686270796639971975929, −15.2515998095181248056063749225, −14.18851445178701934188866041271, −12.63866442149322113668268526096, −11.45527929497832364892729492598, −10.90523464111568847717625873702, −9.72281265927223951612533102595, −8.14207477271956777579927682039, −7.46834270870958329306463337871, −6.19341713073407880977288711317, −5.451405385100108935680192008369, −2.77711195360060724178736773070, −1.79154614289146166293856717680, 0.451248926230370229843089913531, 2.16819056961533872430147658782, 4.283321975225925962425207680659, 5.15536967349438232094523704644, 6.764469685258156481150032004795, 7.90725070899716776711368669206, 9.24581122803209038907294471214, 9.93857225378627214216326657263, 11.03111485755849015109181165288, 11.912213417645264747114121106552, 13.08212276860131143888930249910, 14.86111658180093717899968329883, 15.88083887614264964590285737464, 16.7818274455006955913617617904, 17.42141318638448473913888298959, 18.18174538790862855876354582625, 19.9730881787222749348241980847, 20.442446378786029281954882124452, 21.27714004435215068031540318961, 22.439474919059657003802714929556, 24.0328892740261415061285737602, 24.2796897048310893254839718649, 25.92898040064532886382859887728, 26.66913359558657592675769116630, 27.49944733901012349277216791121

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