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Properties

Label 1-296-296.181-r0-0-0
Degree $1$
Conductor $296$
Sign $0.977 - 0.208i$
Analytic cond. $1.37461$
Root an. cond. $1.37461$
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  + (0.939 − 0.342i)3-s + (−0.173 + 0.984i)5-s + (0.173 − 0.984i)7-s + (0.766 − 0.642i)9-s + (0.5 + 0.866i)11-s + (−0.766 − 0.642i)13-s + (0.173 + 0.984i)15-s + (0.766 − 0.642i)17-s + (0.939 − 0.342i)19-s + (−0.173 − 0.984i)21-s + (−0.5 + 0.866i)23-s + (−0.939 − 0.342i)25-s + (0.5 − 0.866i)27-s + (0.5 + 0.866i)29-s + 31-s + ⋯
L(s)  = 1  + (0.939 − 0.342i)3-s + (−0.173 + 0.984i)5-s + (0.173 − 0.984i)7-s + (0.766 − 0.642i)9-s + (0.5 + 0.866i)11-s + (−0.766 − 0.642i)13-s + (0.173 + 0.984i)15-s + (0.766 − 0.642i)17-s + (0.939 − 0.342i)19-s + (−0.173 − 0.984i)21-s + (−0.5 + 0.866i)23-s + (−0.939 − 0.342i)25-s + (0.5 − 0.866i)27-s + (0.5 + 0.866i)29-s + 31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(296\)    =    \(2^{3} \cdot 37\)
Sign: $0.977 - 0.208i$
Analytic conductor: \(1.37461\)
Root analytic conductor: \(1.37461\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{296} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 296,\ (0:\ ),\ 0.977 - 0.208i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.744122611 - 0.1840241089i\)
\(L(\frac12)\) \(\approx\) \(1.744122611 - 0.1840241089i\)
\(L(1)\) \(\approx\) \(1.440533420 - 0.09004690007i\)
\(L(1)\) \(\approx\) \(1.440533420 - 0.09004690007i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
37 \( 1 \)
good3 \( 1 + (0.939 - 0.342i)T \)
5 \( 1 + (-0.173 + 0.984i)T \)
7 \( 1 + (0.173 - 0.984i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-0.766 - 0.642i)T \)
17 \( 1 + (0.766 - 0.642i)T \)
19 \( 1 + (0.939 - 0.342i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + T \)
41 \( 1 + (0.766 + 0.642i)T \)
43 \( 1 - T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (-0.173 - 0.984i)T \)
59 \( 1 + (-0.173 - 0.984i)T \)
61 \( 1 + (-0.766 - 0.642i)T \)
67 \( 1 + (-0.173 + 0.984i)T \)
71 \( 1 + (-0.939 + 0.342i)T \)
73 \( 1 + T \)
79 \( 1 + (0.173 - 0.984i)T \)
83 \( 1 + (-0.766 + 0.642i)T \)
89 \( 1 + (0.173 + 0.984i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−25.24963385817002031105175143944, −24.53632499792798551896789553863, −24.17615909664330206774255290123, −22.57109059554373974877668803073, −21.436294375696353235224786665585, −21.1759975211480033286195483200, −19.95135887448211807782878925398, −19.28751682025341932438413623872, −18.47325622315987057061527785060, −16.96476027399793153736855268577, −16.251447008883476104203964008382, −15.35871981069190177520643686637, −14.38778294177935894368655849788, −13.615104537150340905296188253226, −12.32777332809022629987007368995, −11.78224472667668770682270655254, −10.13018596654406231865773549136, −9.23209644884284013162348062929, −8.500048449416877380234845797346, −7.76715755795193188757775475939, −6.08428571631666411535212154602, −4.952163930183607773425035398273, −3.95662974543580732934302807197, −2.71433139159036310348245988313, −1.47119384356174972694014747732, 1.34380250178966440887803878282, 2.780232053397429429821409306935, 3.57074993204722112106279574406, 4.82073139033071376002339936166, 6.62876853583092011622444394804, 7.40044784828734358873855156265, 7.92556292610402872936456378511, 9.67687123040277111417360530517, 10.03232235455505345021595848621, 11.44850475800942613742950780081, 12.443444118133571469426218991723, 13.67860982012566133005851235039, 14.30954096778128210027182748636, 15.0091338213085813981632748354, 16.046574284279025811245212273071, 17.56493487845037780785521883913, 18.0391546000602604252442806408, 19.29090370017919515352916064343, 19.89519391999285689684358898325, 20.60299221700550258702467546506, 21.82169821339313480390830307179, 22.81380263077660716438498920709, 23.533965195142658802359400568336, 24.65901915040766332820943937590, 25.45486738169246288911623954995

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