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Properties

Label 1-152-152.5-r0-0-0
Degree $1$
Conductor $152$
Sign $0.977 + 0.211i$
Analytic cond. $0.705885$
Root an. cond. $0.705885$
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 + (−0.5 + 0.866i)7-s + (0.766 + 0.642i)9-s + (0.5 + 0.866i)11-s + (0.939 − 0.342i)13-s + (0.173 − 0.984i)15-s + (0.766 − 0.642i)17-s + (−0.766 + 0.642i)21-s + (0.173 − 0.984i)23-s + (−0.939 + 0.342i)25-s + (0.5 + 0.866i)27-s + (−0.766 − 0.642i)29-s + (−0.5 + 0.866i)31-s + (0.173 + 0.984i)33-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)3-s + (−0.173 − 0.984i)5-s + (−0.5 + 0.866i)7-s + (0.766 + 0.642i)9-s + (0.5 + 0.866i)11-s + (0.939 − 0.342i)13-s + (0.173 − 0.984i)15-s + (0.766 − 0.642i)17-s + (−0.766 + 0.642i)21-s + (0.173 − 0.984i)23-s + (−0.939 + 0.342i)25-s + (0.5 + 0.866i)27-s + (−0.766 − 0.642i)29-s + (−0.5 + 0.866i)31-s + (0.173 + 0.984i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $0.977 + 0.211i$
Analytic conductor: \(0.705885\)
Root analytic conductor: \(0.705885\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{152} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 152,\ (0:\ ),\ 0.977 + 0.211i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.451355777 + 0.1555277941i\)
\(L(\frac12)\) \(\approx\) \(1.451355777 + 0.1555277941i\)
\(L(1)\) \(\approx\) \(1.346318278 + 0.08944153090i\)
\(L(1)\) \(\approx\) \(1.346318278 + 0.08944153090i\)

Euler product

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

−27.75526928570913589453058901056, −26.72909893637471040555321843973, −26.04304796236316607533117177592, −25.399169575559338034132003191343, −23.96359027868957601929000981318, −23.30465278323595424027244634195, −22.05516866745365144723332399048, −21.03523976341117807012526978177, −19.87828069591591168658324678414, −19.093648367133613346197182622276, −18.43688842502740625541941109852, −16.98554764496508915289627614508, −15.807157828889224332622784791600, −14.66571880201762653101760920114, −13.874928875890808543371123199635, −13.09444677959908444440228416269, −11.52469344823527651654986992954, −10.45431931789208719326393070013, −9.32058836841574133646943040389, −8.05861147388794715078001524178, −7.061864242303558316167897678727, −6.128720624219655683931681079523, −3.69286691783667600779319540309, −3.38608480676132664443958577, −1.5328170782760092131056298850, 1.69084118533216752300594774702, 3.17918642631777690129848488282, 4.39057771877197107094704153359, 5.61989047200989797772061512625, 7.295605190633353972011561984919, 8.618922345802028364731773395561, 9.14615068640587271261767806404, 10.260928388372601720176602280106, 12.01743810227987078437765430772, 12.79042025493666316993335390223, 13.88964020677666645130941533376, 15.150384269432742810070750082290, 15.848251305944933868619167752342, 16.81864057175114686473436090258, 18.35683445636565130183571559652, 19.26250526207942663166390748740, 20.40075256779532889804453713472, 20.81115880460154650312522615552, 22.072960076318831557331153079232, 23.12353286585875422411171036704, 24.523697983405369882012576857626, 25.230394955516611433938207546299, 25.82055633703427888882345059372, 27.26644722665516554627093976256, 27.96729047747921506341599723302

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