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Properties

Label 1-296-296.35-r0-0-0
Degree $1$
Conductor $296$
Sign $-0.566 + 0.823i$
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  + (−0.173 − 0.984i)3-s + (−0.642 − 0.766i)5-s + (−0.766 + 0.642i)7-s + (−0.939 + 0.342i)9-s + (0.5 − 0.866i)11-s + (−0.342 + 0.939i)13-s + (−0.642 + 0.766i)15-s + (−0.342 − 0.939i)17-s + (−0.984 + 0.173i)19-s + (0.766 + 0.642i)21-s + (−0.866 + 0.5i)23-s + (−0.173 + 0.984i)25-s + (0.5 + 0.866i)27-s + (−0.866 − 0.5i)29-s − i·31-s + ⋯
L(s)  = 1  + (−0.173 − 0.984i)3-s + (−0.642 − 0.766i)5-s + (−0.766 + 0.642i)7-s + (−0.939 + 0.342i)9-s + (0.5 − 0.866i)11-s + (−0.342 + 0.939i)13-s + (−0.642 + 0.766i)15-s + (−0.342 − 0.939i)17-s + (−0.984 + 0.173i)19-s + (0.766 + 0.642i)21-s + (−0.866 + 0.5i)23-s + (−0.173 + 0.984i)25-s + (0.5 + 0.866i)27-s + (−0.866 − 0.5i)29-s − i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.566 + 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.566 + 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(296\)    =    \(2^{3} \cdot 37\)
Sign: $-0.566 + 0.823i$
Analytic conductor: \(1.37461\)
Root analytic conductor: \(1.37461\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{296} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 296,\ (0:\ ),\ -0.566 + 0.823i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01954622060 - 0.03717487922i\)
\(L(\frac12)\) \(\approx\) \(0.01954622060 - 0.03717487922i\)
\(L(1)\) \(\approx\) \(0.5506323201 - 0.2164160114i\)
\(L(1)\) \(\approx\) \(0.5506323201 - 0.2164160114i\)

Euler product

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

−26.062459739716414658278795820514, −25.553427449322851922951943516374, −23.94791513928024360902855426394, −23.01286249379431321050776451041, −22.4446663683334438157022131234, −21.82779946023702520943262537006, −20.30863483086680823198647192882, −19.95881498129444500920419351629, −18.92573732838161903309374439470, −17.55589142363980978044490551355, −16.91460303887396873778256068666, −15.783136261183085420650287557176, −15.07586189790149054213805414196, −14.426699371072350666127788287, −12.95382198124834965195019403777, −11.98862452130392807170776187527, −10.682883012991547177469718288, −10.36459316823187020508286031746, −9.26027115035750246093306255466, −7.96319848460839927508519564533, −6.84767500364608771922737664524, −5.881474489629264502269450495139, −4.25766811961533729603986320040, −3.801326763792955643452677679167, −2.51380049402269063936861885379, 0.027134008546020710181920507942, 1.628149750084993083727447686565, 2.96874866802238485476853683956, 4.33985273561316449175837486614, 5.710963715416412373882564604577, 6.54673904721604695335692560128, 7.65569393421656156757346840120, 8.71804160979296144237861454662, 9.3811678323492026028328337679, 11.23438406219350400336871682359, 11.88510401783679848696238415890, 12.6611494903669797167609050365, 13.53625758794876761819831283864, 14.55676416678312783859370533470, 15.98560081176589981386224498027, 16.499855567575867979212719201536, 17.536683276148930585132660850, 18.77905285122110334257276290700, 19.28379343211733728050305309981, 19.96609047161738916691730194547, 21.282739606179463571841020472649, 22.28853895961040458789441447681, 23.18883237100886963545750000281, 24.0769398730112609134344246614, 24.65720350664136266012643800034

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