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Properties

Label 1-320-320.93-r1-0-0
Degree $1$
Conductor $320$
Sign $0.864 + 0.502i$
Analytic cond. $34.3887$
Root an. cond. $34.3887$
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  + (−0.382 + 0.923i)3-s + (−0.707 − 0.707i)7-s + (−0.707 − 0.707i)9-s + (−0.923 + 0.382i)11-s + (−0.923 − 0.382i)13-s + 17-s + (−0.382 + 0.923i)19-s + (0.923 − 0.382i)21-s + (0.707 − 0.707i)23-s + (0.923 − 0.382i)27-s + (0.923 + 0.382i)29-s − 31-s − i·33-s + (0.923 − 0.382i)37-s + (0.707 − 0.707i)39-s + ⋯
L(s)  = 1  + (−0.382 + 0.923i)3-s + (−0.707 − 0.707i)7-s + (−0.707 − 0.707i)9-s + (−0.923 + 0.382i)11-s + (−0.923 − 0.382i)13-s + 17-s + (−0.382 + 0.923i)19-s + (0.923 − 0.382i)21-s + (0.707 − 0.707i)23-s + (0.923 − 0.382i)27-s + (0.923 + 0.382i)29-s − 31-s − i·33-s + (0.923 − 0.382i)37-s + (0.707 − 0.707i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.864 + 0.502i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.864 + 0.502i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.864 + 0.502i$
Analytic conductor: \(34.3887\)
Root analytic conductor: \(34.3887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 320,\ (1:\ ),\ 0.864 + 0.502i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.047349513 + 0.2821801561i\)
\(L(\frac12)\) \(\approx\) \(1.047349513 + 0.2821801561i\)
\(L(1)\) \(\approx\) \(0.7782993151 + 0.1571119424i\)
\(L(1)\) \(\approx\) \(0.7782993151 + 0.1571119424i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (-0.382 + 0.923i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
11 \( 1 + (-0.923 + 0.382i)T \)
13 \( 1 + (-0.923 - 0.382i)T \)
17 \( 1 + T \)
19 \( 1 + (-0.382 + 0.923i)T \)
23 \( 1 + (0.707 - 0.707i)T \)
29 \( 1 + (0.923 + 0.382i)T \)
31 \( 1 - T \)
37 \( 1 + (0.923 - 0.382i)T \)
41 \( 1 + (-0.707 - 0.707i)T \)
43 \( 1 + (0.382 + 0.923i)T \)
47 \( 1 + T \)
53 \( 1 + (-0.382 - 0.923i)T \)
59 \( 1 + (0.382 + 0.923i)T \)
61 \( 1 + (0.923 + 0.382i)T \)
67 \( 1 + (0.382 - 0.923i)T \)
71 \( 1 + (0.707 - 0.707i)T \)
73 \( 1 + (-0.707 + 0.707i)T \)
79 \( 1 - iT \)
83 \( 1 + (0.923 + 0.382i)T \)
89 \( 1 + (0.707 - 0.707i)T \)
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

−24.96918375725828639725553085679, −23.67265437310847004608091003928, −23.45242325530493965337370162533, −22.07504783349825516428196063070, −21.60653135440798158199164557548, −20.18366573213831745582550780018, −19.05206345914615432723821895559, −18.85526301535669778566342244956, −17.68089015947059012792529760622, −16.818769376264044715416179492576, −15.88088670601464824602616813112, −14.81343252579956272270507236601, −13.6242699917686458646236110688, −12.84639755323340469151821474327, −12.09675731298773373889884069803, −11.15988434594816495006095349604, −9.96408182079964735686864654509, −8.847165117666575676750791125830, −7.738193130094628099856725958147, −6.85583721387285284382572770049, −5.78860129277877382481338543957, −4.99132507449871882493374439436, −3.071288682276864931822153076432, −2.243088391219961304489650736682, −0.61366422405991129820852737841, 0.601214249101161894107409681458, 2.68926075704009272992453821269, 3.72370012231440346772997324078, 4.82950403067264927780520390157, 5.73561115723882826410733903495, 6.98347106107434711748094251862, 8.06414233049729799035687470462, 9.45523818390352341231535532487, 10.22499025201800254198800679075, 10.74141402771429501624317127838, 12.2143687290012302266510294265, 12.869393965021040709402070462700, 14.30081189446801727818805665362, 15.00649519205030943254269547906, 16.142503693372952660322513587372, 16.67905341287308509747575109321, 17.57181143967594204602221317007, 18.70925889552018295306743120920, 19.828582002509913794780440482369, 20.62906220308126508806557126528, 21.38548202344404089674430151605, 22.45179712549133179382761061740, 23.08992650291743684841394285218, 23.77512870111215838228206153801, 25.29502015752552956321350805391

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