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Properties

Label 1-152-152.125-r0-0-0
Degree $1$
Conductor $152$
Sign $0.305 + 0.952i$
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  + (0.5 + 0.866i)3-s + (0.5 + 0.866i)5-s + 7-s + (−0.5 + 0.866i)9-s − 11-s + (0.5 − 0.866i)13-s + (−0.5 + 0.866i)15-s + (−0.5 − 0.866i)17-s + (0.5 + 0.866i)21-s + (−0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s − 27-s + (0.5 − 0.866i)29-s + 31-s + (−0.5 − 0.866i)33-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s + (0.5 + 0.866i)5-s + 7-s + (−0.5 + 0.866i)9-s − 11-s + (0.5 − 0.866i)13-s + (−0.5 + 0.866i)15-s + (−0.5 − 0.866i)17-s + (0.5 + 0.866i)21-s + (−0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s − 27-s + (0.5 − 0.866i)29-s + 31-s + (−0.5 − 0.866i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $0.305 + 0.952i$
Analytic conductor: \(0.705885\)
Root analytic conductor: \(0.705885\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{152} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 152,\ (0:\ ),\ 0.305 + 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.147613260 + 0.8367863632i\)
\(L(\frac12)\) \(\approx\) \(1.147613260 + 0.8367863632i\)
\(L(1)\) \(\approx\) \(1.211919894 + 0.5141743022i\)
\(L(1)\) \(\approx\) \(1.211919894 + 0.5141743022i\)

Euler product

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

−28.181691210310900214309642828645, −26.60926855801032402409909195848, −25.82945786005002831533725021085, −24.717486202765027203318882516308, −24.048291440503326436252159732718, −23.43641958389564792851957121158, −21.592710716759199292953098322472, −20.83406731447698445255615101190, −20.07210641149874769242173895603, −18.77447526500816632179759277411, −17.93260579528068293003866776237, −17.08682065198353230288185570066, −15.75110982204238483438838752028, −14.422606857360046269742069211028, −13.6026449623473760098104632157, −12.70862975941474851588592905707, −11.68369464139208154010639137733, −10.278200454641890155587584381573, −8.6130762907397055136784357360, −8.358381514448627402772700819368, −6.83891003504922033180580520024, −5.56252200396820355207073402771, −4.28450673455783690557507279492, −2.354404836100852583661698175023, −1.376028657961119821359681902191, 2.20318121968817272940085597248, 3.237453585557295561336128453018, 4.75897476599268208142667395651, 5.768141129351088683572439157424, 7.52058287249586169618614448605, 8.43378255532778983789734282390, 9.8402406208402311978208272347, 10.62656235165231773823627995797, 11.49452421307100383737739105818, 13.45277477990436370220448055502, 14.0830356523031143139866491746, 15.25428191237479321296413166086, 15.80149202187477585205065727863, 17.4812161356489695112924428464, 18.10730883529562467477813885197, 19.37354172534408857467242521214, 20.76071198305173055716065788470, 21.08409786933521654900929089796, 22.25423048503208868369239988758, 23.09067990715521247644768797178, 24.53135187335190471526264574378, 25.51996734130746424557358993655, 26.326054586465384311583537552458, 27.144531141401224401344655701292, 27.98691435014713718116851355243

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