Properties

Degree 1
Conductor $ 2^{6} \cdot 5 $
Sign $-0.471 - 0.881i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.923 − 0.382i)3-s + (−0.707 + 0.707i)7-s + (0.707 + 0.707i)9-s + (−0.923 + 0.382i)11-s + (0.382 − 0.923i)13-s i·17-s + (0.382 − 0.923i)19-s + (0.923 − 0.382i)21-s + (0.707 + 0.707i)23-s + (−0.382 − 0.923i)27-s + (−0.923 − 0.382i)29-s − 31-s + 33-s + (−0.382 − 0.923i)37-s + (−0.707 + 0.707i)39-s + ⋯
L(s,χ)  = 1  + (−0.923 − 0.382i)3-s + (−0.707 + 0.707i)7-s + (0.707 + 0.707i)9-s + (−0.923 + 0.382i)11-s + (0.382 − 0.923i)13-s i·17-s + (0.382 − 0.923i)19-s + (0.923 − 0.382i)21-s + (0.707 + 0.707i)23-s + (−0.382 − 0.923i)27-s + (−0.923 − 0.382i)29-s − 31-s + 33-s + (−0.382 − 0.923i)37-s + (−0.707 + 0.707i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.471 - 0.881i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 320 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.471 - 0.881i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(320\)    =    \(2^{6} \cdot 5\)
\( \varepsilon \)  =  $-0.471 - 0.881i$
motivic weight  =  \(0\)
character  :  $\chi_{320} (29, \cdot )$
Sato-Tate  :  $\mu(16)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 320,\ (0:\ ),\ -0.471 - 0.881i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.2389457912 - 0.3986569682i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.2389457912 - 0.3986569682i\)
\(L(\chi,1)\)  \(\approx\)  \(0.6114101318 - 0.1392262160i\)
\(L(1,\chi)\)  \(\approx\)  \(0.6114101318 - 0.1392262160i\)

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−25.67205338369464264878437064867, −24.15723517584287515554221780305, −23.5968391723227032868271820056, −22.83661440576424708470515011246, −21.93712075911978731582046321282, −21.08327639514474995868956273896, −20.26022727281397190777989290907, −18.8796319308629023450541516704, −18.37803318712516363985054455107, −16.93287564670275112423301319577, −16.60493753523726689414429524989, −15.7170737901769335001580243505, −14.60928903679419848143662419200, −13.32470743531318833431352097533, −12.64369541053354467819142772494, −11.4484826017233090477469564605, −10.58411737312218546374066569476, −9.92977874314145081042421897984, −8.696074137695012463753242554280, −7.30160820240323199658255608615, −6.37750182431140076089900200377, −5.4613010328467274225022648296, −4.23446736249077300031069345683, −3.332731766982684298258408314009, −1.408774017011029327123254090993, 0.344138418981639434008158697, 2.12704980602545944467143945787, 3.33421709998110100591216473895, 5.14859997600436576703203748703, 5.53520240140147990491325936898, 6.86223004660017830959673390243, 7.62343282425388475495184972688, 9.0504652400876238479677011990, 10.076309287156288800083675651089, 11.07086588344450410404042017538, 11.954610913143270630753764447755, 12.99998546539954373250336512538, 13.389322998672437725741503933148, 15.21779033330284809733204647512, 15.7794988056277333944558001246, 16.69107933611787110688187811017, 17.92439364542898107465654308163, 18.28889795965469394658942049637, 19.296277231675683439277556252643, 20.392663165735848106227026587473, 21.47916406146385012578686261951, 22.40437515043406319623537827943, 22.971335031507894182463017092853, 23.83460570726596911681086326917, 24.89098632704483464856222100134

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