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Properties

Label 1-160-160.107-r0-0-0
Degree $1$
Conductor $160$
Sign $0.731 + 0.681i$
Analytic cond. $0.743036$
Root an. cond. $0.743036$
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

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

Invariants

Degree: \(1\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $0.731 + 0.681i$
Analytic conductor: \(0.743036\)
Root analytic conductor: \(0.743036\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 160,\ (0:\ ),\ 0.731 + 0.681i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.373390420 + 0.5405426148i\)
\(L(\frac12)\) \(\approx\) \(1.373390420 + 0.5405426148i\)
\(L(1)\) \(\approx\) \(1.311749864 + 0.3226931739i\)
\(L(1)\) \(\approx\) \(1.311749864 + 0.3226931739i\)

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

−27.74832533145168024057360511624, −26.54426834630535248256845001494, −25.66593441507393264676373581195, −24.87947648380443170523183426257, −23.769752006617529397040548481615, −23.25686298664791538188637444016, −21.58365129300833243507570585993, −20.686549719013125451511833998516, −19.96138117519833669859325603632, −18.7087813439030914889426900954, −17.97386187146812465861510625436, −17.06658049893386900245860743273, −15.131948277979182807460424709399, −14.965101202929778871363091685101, −13.38743785429130440237378440831, −12.85564291335614844298800613144, −11.474305460552340463316836076055, −10.354060812679483925785018998459, −8.821807244816830170346765652071, −8.04479245910168128629063794624, −7.06427700218216386105059657722, −5.621747340683315268415916036587, −4.15956019814203188672818291818, −2.64078125506603360663881000489, −1.45551669781569067668430145632, 1.86989227407203050961337783450, 3.27491695106231472760842870270, 4.5130335109353636723447890539, 5.57071356244337538245356796326, 7.39327658416458129404101484661, 8.443870921823503765793795561329, 9.24990424937140509702577272431, 10.72203260826857643204310751229, 11.30321917800023590694340465335, 13.04371374114541929815673873025, 14.08546876060431375567921323099, 14.7976451891059254796283634839, 15.98467500354679008899254213691, 16.74036462187176419463528303824, 18.263498203844144910749220839903, 19.02014586706438601002348662957, 20.41088506162179621984324481873, 21.03211728078970417472991862623, 21.71931529536307694776679142967, 23.10928032126328013809405510457, 24.13541143943284678070717982897, 25.14101833560387645856318870300, 26.074897715986770025801632886016, 27.05666919363958839343277534156, 27.5936166783575534076628358191

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