Properties

Label 1-160-160.13-r1-0-0
Degree $1$
Conductor $160$
Sign $-0.731 + 0.681i$
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 − 7-s i·9-s + (−0.707 + 0.707i)11-s + (−0.707 + 0.707i)13-s i·17-s + (−0.707 − 0.707i)19-s + (−0.707 + 0.707i)21-s − 23-s + (−0.707 − 0.707i)27-s + (0.707 + 0.707i)29-s + 31-s + i·33-s + (−0.707 − 0.707i)37-s i·39-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)3-s − 7-s i·9-s + (−0.707 + 0.707i)11-s + (−0.707 + 0.707i)13-s i·17-s + (−0.707 − 0.707i)19-s + (−0.707 + 0.707i)21-s − 23-s + (−0.707 − 0.707i)27-s + (0.707 + 0.707i)29-s + 31-s + i·33-s + (−0.707 − 0.707i)37-s i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.731 + 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.731 + 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $-0.731 + 0.681i$
Analytic conductor: \(17.1943\)
Root analytic conductor: \(17.1943\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 160,\ (1:\ ),\ -0.731 + 0.681i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07981111830 + 0.2027810986i\)
\(L(\frac12)\) \(\approx\) \(0.07981111830 + 0.2027810986i\)
\(L(1)\) \(\approx\) \(0.8525966743 - 0.1153223352i\)
\(L(1)\) \(\approx\) \(0.8525966743 - 0.1153223352i\)

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 \( 1 + T \)
11 \( 1 + (0.707 - 0.707i)T \)
13 \( 1 + (0.707 - 0.707i)T \)
17 \( 1 + iT \)
19 \( 1 + (0.707 + 0.707i)T \)
23 \( 1 + T \)
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 \( 1 + iT \)
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 \( 1 + T \)
79 \( 1 - T \)
83 \( 1 + (0.707 - 0.707i)T \)
89 \( 1 + iT \)
97 \( 1 + iT \)
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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

−27.00972916791462930949821280038, −26.32531053732845755749464608221, −25.37046998877987155430647117335, −24.577647420998334614833993940349, −23.09278704082799075451964984365, −22.2506420608529252562547043305, −21.29970105610677630641133663170, −20.33242701366020686449763109767, −19.43020908248412844892154695984, −18.58219029173022952672851272741, −17.0296194002522163493892431228, −16.00312055264628369194619061492, −15.40451114382543454720007666738, −14.07827020741281670653368058224, −13.2872211153691236372513186372, −12.03390716666866944228928613135, −10.405853244399514859220775417224, −9.901678084619627211908837742444, −8.61918531455775110826864475423, −7.64032432721772855233994811884, −6.06908410664594062037771826032, −4.78240766195343510451581709691, −3.386353355316583829596316506900, −2.52362158153559086012638082101, −0.066890065052554111909579382640, 1.90195784451822220503839948331, 2.99316453917975398799962947474, 4.41731248953125935014595507156, 6.23959132729753004492265230181, 7.08401167405166132094105823294, 8.25240995924161066413763558259, 9.390359037249696246388605697141, 10.36162736207258187583251678414, 12.14059223357293293452050151277, 12.7836546150922194643640382561, 13.76853012472557146032850779299, 14.887471025382976103316055706162, 15.8233573929068248377590275548, 17.1830058094368382858030632213, 18.184725529964373656718175886652, 19.35942886251866868238889424285, 19.72181325441686839396809855573, 21.00886480724344850296223031320, 22.05728171973624659583484410913, 23.36334066538258421423624676875, 23.98849091296132890140007520130, 25.1959587132099039680611108936, 26.017681649864817800649854145228, 26.51083105301822997359450854896, 28.19439225610654939347740276056

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