Properties

Label 1-160-160.69-r0-0-0
Degree $1$
Conductor $160$
Sign $0.195 - 0.980i$
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 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

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

Invariants

Degree: \(1\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $0.195 - 0.980i$
Analytic conductor: \(0.743036\)
Root analytic conductor: \(0.743036\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 160,\ (0:\ ),\ 0.195 - 0.980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.012196078 - 0.8306878539i\)
\(L(\frac12)\) \(\approx\) \(1.012196078 - 0.8306878539i\)
\(L(1)\) \(\approx\) \(1.135469945 - 0.4758960435i\)
\(L(1)\) \(\approx\) \(1.135469945 - 0.4758960435i\)

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

−27.95946221726042572087772920620, −27.04858321510119738759487237801, −26.064613798052850741516107760345, −25.180728004007236795631248328155, −24.50552146168699241581336592249, −22.86602189390774240021231657292, −22.16169797081766783796969281739, −20.985398697401483435002876722171, −20.457562033864123864585357685380, −19.16576151411213642429796838296, −18.35351926037401328227125204390, −16.99027563590208659303975324139, −15.796763450715809362495590090490, −15.10229687672500820697130956248, −14.24418310968094429103582483913, −12.865306080048940006667501025443, −11.910697402956834488931902788569, −10.29343770845275946087303644508, −9.70603746314394862866250275147, −8.397786483790324300422083170045, −7.57089678592247254108672915513, −5.66805275080819087226729479195, −4.75613378542134487825556676355, −3.19049471852020818030556563483, −2.2251930829374362149409651011, 1.10974534005244364608051342893, 2.71654762822015209250158555261, 3.86795287692737029764308027793, 5.51352677808585659908818825440, 7.102462555135125364841203903765, 7.63987423781880119952524994967, 8.97152242300968380276267828888, 10.078556487330709962861068584769, 11.41629291958061080396901897919, 12.5931813094584955740937440065, 13.72755668278879010089673610519, 14.16997784201698278581356748280, 15.55694200585135413971801776085, 16.764634092876781599780969705816, 17.785963232085293620188328051, 18.93514395386250023022351995449, 19.61469634331516042875781823606, 20.64658457420551718603030707848, 21.51763772327470678631652987027, 23.0225732492869939621815938218, 23.9216287770445373587320393078, 24.48173768606164718419609819812, 25.91511984932619122112676081883, 26.35198176595689224335683933968, 27.38257594686289597465329591640

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