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Properties

Label 1-160-160.19-r1-0-0
Degree $1$
Conductor $160$
Sign $-0.980 + 0.195i$
Analytic cond. $17.1943$
Root an. cond. $17.1943$
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  + (−0.707 − 0.707i)3-s − i·7-s + i·9-s + (0.707 − 0.707i)11-s + (−0.707 − 0.707i)13-s + 17-s + (−0.707 − 0.707i)19-s + (−0.707 + 0.707i)21-s − i·23-s + (0.707 − 0.707i)27-s + (−0.707 − 0.707i)29-s − 31-s − 33-s + (−0.707 + 0.707i)37-s + i·39-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)3-s − i·7-s + i·9-s + (0.707 − 0.707i)11-s + (−0.707 − 0.707i)13-s + 17-s + (−0.707 − 0.707i)19-s + (−0.707 + 0.707i)21-s − i·23-s + (0.707 − 0.707i)27-s + (−0.707 − 0.707i)29-s − 31-s − 33-s + (−0.707 + 0.707i)37-s + i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.980 + 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.980 + 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $-0.980 + 0.195i$
Analytic conductor: \(17.1943\)
Root analytic conductor: \(17.1943\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 160,\ (1:\ ),\ -0.980 + 0.195i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.05961928267 - 0.6053247353i\)
\(L(\frac12)\) \(\approx\) \(-0.05961928267 - 0.6053247353i\)
\(L(1)\) \(\approx\) \(0.6488432529 - 0.3388459284i\)
\(L(1)\) \(\approx\) \(0.6488432529 - 0.3388459284i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (-0.707 - 0.707i)T \)
7 \( 1 - iT \)
11 \( 1 + (0.707 - 0.707i)T \)
13 \( 1 + (-0.707 - 0.707i)T \)
17 \( 1 + T \)
19 \( 1 + (-0.707 - 0.707i)T \)
23 \( 1 - iT \)
29 \( 1 + (-0.707 - 0.707i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.707 + 0.707i)T \)
41 \( 1 + iT \)
43 \( 1 + (-0.707 + 0.707i)T \)
47 \( 1 - T \)
53 \( 1 + (0.707 - 0.707i)T \)
59 \( 1 + (-0.707 + 0.707i)T \)
61 \( 1 + (-0.707 - 0.707i)T \)
67 \( 1 + (-0.707 - 0.707i)T \)
71 \( 1 + iT \)
73 \( 1 - iT \)
79 \( 1 + T \)
83 \( 1 + (0.707 + 0.707i)T \)
89 \( 1 - iT \)
97 \( 1 - T \)
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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

−27.90485823391598115449264664510, −27.47221027379942658178754904931, −26.20114295598827070115697030367, −25.23274973010978728034375370702, −24.15269358718013574786403675907, −23.014598284524776789976571185993, −22.199008680865757892550631624345, −21.43012996451709752183814860596, −20.47731913841692254083767240582, −19.10071928676365515428106446788, −18.159926842879871104622355033449, −16.97258396039702561022135533620, −16.3408627747313864407206155216, −14.97542402002371519137617758357, −14.518981017988854234565572206626, −12.3681495369906258581594143763, −12.105756805283033730665942218914, −10.72922275260190127367444832683, −9.63840740111949047220491365109, −8.807086606388653281120072983408, −7.06214727448325131029271936277, −5.89724361031660043718183623060, −4.87562073122237897021052747595, −3.66666023450097470519411040879, −1.901695843343183805562260372406, 0.25030646256263630909496929773, 1.4944189295840443446821227095, 3.35545499275294582615079749758, 4.889678686582969094522236047306, 6.08544160042652755916360793709, 7.186651836669783391508055818242, 8.06225174121743697550259797676, 9.750998152721211185895078914699, 10.87693923347850058356153469166, 11.74614745038328772736113089452, 12.94413522312942921658019038562, 13.73708037930455791706648368841, 14.93018857550020635404468394333, 16.561048269123054989118743945871, 17.034262881854819456116393295, 18.01436069616856215979418089208, 19.28222115007761844336834373094, 19.85225411076000277362636279158, 21.34310914614512441258166404494, 22.36591924519099324982214609230, 23.25382534809808164209555474488, 24.04806558220522528229107881720, 24.929927182745032254515999330023, 26.03678095997107185080899339984, 27.34134836667571011641688343854

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