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Properties

Label 1-320-320.197-r1-0-0
Degree $1$
Conductor $320$
Sign $0.256 - 0.966i$
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  + (0.923 − 0.382i)3-s + (0.707 − 0.707i)7-s + (0.707 − 0.707i)9-s + (−0.382 + 0.923i)11-s + (−0.382 − 0.923i)13-s + 17-s + (0.923 − 0.382i)19-s + (0.382 − 0.923i)21-s + (−0.707 − 0.707i)23-s + (0.382 − 0.923i)27-s + (0.382 + 0.923i)29-s − 31-s − i·33-s + (0.382 − 0.923i)37-s + (−0.707 − 0.707i)39-s + ⋯
L(s)  = 1  + (0.923 − 0.382i)3-s + (0.707 − 0.707i)7-s + (0.707 − 0.707i)9-s + (−0.382 + 0.923i)11-s + (−0.382 − 0.923i)13-s + 17-s + (0.923 − 0.382i)19-s + (0.382 − 0.923i)21-s + (−0.707 − 0.707i)23-s + (0.382 − 0.923i)27-s + (0.382 + 0.923i)29-s − 31-s − i·33-s + (0.382 − 0.923i)37-s + (−0.707 − 0.707i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.256 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.256 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.256 - 0.966i$
Analytic conductor: \(34.3887\)
Root analytic conductor: \(34.3887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 320,\ (1:\ ),\ 0.256 - 0.966i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.353004410 - 1.810668326i\)
\(L(\frac12)\) \(\approx\) \(2.353004410 - 1.810668326i\)
\(L(1)\) \(\approx\) \(1.542072531 - 0.4897536869i\)
\(L(1)\) \(\approx\) \(1.542072531 - 0.4897536869i\)

Euler product

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

−25.07646940154810375142206429622, −24.38003217428209527498910419169, −23.559919603750857097235769641653, −22.02732050933312361131443644971, −21.47458220584506653014302079336, −20.808888954610015510079262650408, −19.74346857856836232844269741654, −18.82003783736376110862945866197, −18.23943694268577234618590541574, −16.7357013631446203073420014584, −15.984745789991983162431622231, −14.996613455165563439373892636346, −14.19399854962656515373004948741, −13.53309178393370193246135026874, −12.12979655853172682490088247336, −11.30527957932846797896203881589, −10.01377896411497748022269200039, −9.22240638991943455904141545613, −8.18889024054933303453151503562, −7.56385214689060739037498539214, −5.8911588701052687549154010288, −4.901635267507081963952320489897, −3.67507703977414142176478910819, −2.62402855688201564880078525400, −1.46992507325797550572236741206, 0.81057382519276241432610798756, 2.031069325935970473513219627374, 3.19726922931442380817718988187, 4.35947161826158482163007829848, 5.516044302319242904736571255386, 7.31276212117774272670583662980, 7.51598659038259846028276414219, 8.66895334234991863554968467033, 9.86731163809203375285693201760, 10.590938085856375877037830423769, 12.11033683817177539049012450665, 12.81256291757223620353931985511, 13.92900232372079574167802235563, 14.5718275473629087774081133116, 15.412732528406807654951787346421, 16.59870800499848652652877442853, 17.91548446353550211070909794398, 18.19836945150911371969889293542, 19.62664051128251541843645491140, 20.29177334607307790559259154247, 20.79624145154979429178133776417, 22.00004230020345882218801945660, 23.19067288802903113191743074764, 23.90556733283297886231195681332, 24.81090170014263475193133752146

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