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Properties

Label 1-344-344.331-r0-0-0
Degree $1$
Conductor $344$
Sign $0.954 + 0.298i$
Analytic cond. $1.59752$
Root an. cond. $1.59752$
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  + (−0.0747 + 0.997i)3-s + (0.955 + 0.294i)5-s + (−0.5 − 0.866i)7-s + (−0.988 − 0.149i)9-s + (0.623 − 0.781i)11-s + (0.733 − 0.680i)13-s + (−0.365 + 0.930i)15-s + (0.955 − 0.294i)17-s + (0.988 − 0.149i)19-s + (0.900 − 0.433i)21-s + (−0.365 − 0.930i)23-s + (0.826 + 0.563i)25-s + (0.222 − 0.974i)27-s + (0.0747 + 0.997i)29-s + (−0.826 + 0.563i)31-s + ⋯
L(s)  = 1  + (−0.0747 + 0.997i)3-s + (0.955 + 0.294i)5-s + (−0.5 − 0.866i)7-s + (−0.988 − 0.149i)9-s + (0.623 − 0.781i)11-s + (0.733 − 0.680i)13-s + (−0.365 + 0.930i)15-s + (0.955 − 0.294i)17-s + (0.988 − 0.149i)19-s + (0.900 − 0.433i)21-s + (−0.365 − 0.930i)23-s + (0.826 + 0.563i)25-s + (0.222 − 0.974i)27-s + (0.0747 + 0.997i)29-s + (−0.826 + 0.563i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.954 + 0.298i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.954 + 0.298i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(344\)    =    \(2^{3} \cdot 43\)
Sign: $0.954 + 0.298i$
Analytic conductor: \(1.59752\)
Root analytic conductor: \(1.59752\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{344} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 344,\ (0:\ ),\ 0.954 + 0.298i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.447375912 + 0.2210374163i\)
\(L(\frac12)\) \(\approx\) \(1.447375912 + 0.2210374163i\)
\(L(1)\) \(\approx\) \(1.195761356 + 0.1929363124i\)
\(L(1)\) \(\approx\) \(1.195761356 + 0.1929363124i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
43 \( 1 \)
good3 \( 1 + (-0.0747 + 0.997i)T \)
5 \( 1 + (0.955 + 0.294i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (0.623 - 0.781i)T \)
13 \( 1 + (0.733 - 0.680i)T \)
17 \( 1 + (0.955 - 0.294i)T \)
19 \( 1 + (0.988 - 0.149i)T \)
23 \( 1 + (-0.365 - 0.930i)T \)
29 \( 1 + (0.0747 + 0.997i)T \)
31 \( 1 + (-0.826 + 0.563i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.900 - 0.433i)T \)
47 \( 1 + (-0.623 - 0.781i)T \)
53 \( 1 + (0.733 + 0.680i)T \)
59 \( 1 + (-0.222 + 0.974i)T \)
61 \( 1 + (0.826 + 0.563i)T \)
67 \( 1 + (-0.988 + 0.149i)T \)
71 \( 1 + (0.365 - 0.930i)T \)
73 \( 1 + (0.733 - 0.680i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.0747 - 0.997i)T \)
89 \( 1 + (-0.0747 + 0.997i)T \)
97 \( 1 + (0.623 - 0.781i)T \)
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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

−24.97180600415433818568666990836, −24.12752795966407823379185318586, −23.08790275966695862122141736471, −22.290908935961466190322222387909, −21.35039895500796238541649060241, −20.385637449493451023849593786083, −19.36057600767593466046523716806, −18.540709132274480325206238925799, −17.83600270921324378547584678077, −16.977659734242565368707794955763, −16.03986237242271926825810075648, −14.68387246340283414737464008013, −13.86191794744265262179738397214, −13.01907545642406239572858400313, −12.191128459851493241010482928643, −11.46440452573703854340022925128, −9.80228746814631223078431485582, −9.2230543589980461639950709837, −8.09241527447165459073707207947, −6.879210742623271067850591042771, −6.01248754526762728172305331204, −5.333315142537510646700574911726, −3.531682953101222879380901380763, −2.14968665673179404311145807916, −1.40914874407160177068248223671, 1.1068442738854476862032213345, 3.08396337537191678488217466738, 3.6043036563627622176663214711, 5.11848409437135881016824877468, 5.94205634613343290199972434650, 6.95335300576500031539029541188, 8.47857504680628514346497080583, 9.414551741037869618071346759255, 10.319187203316753002020944145206, 10.785840761643026849222566188320, 12.06626295272154563091588076537, 13.515230348347876133067958056340, 14.02665362265441395195714685057, 14.94790597987851237748711987829, 16.390320696008680495655764317922, 16.51475402375444093246641382078, 17.69054879337779040739761475865, 18.59079296371144981832517274640, 19.94494303607552994059279872088, 20.54632432169197168529203077565, 21.452445640969313224683449549947, 22.33776653693932979676437133056, 22.797433062044103313163625702766, 23.99256260312223880756975622105, 25.27677171537494093426453922652

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