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Properties

Label 2-148-148.123-c0-0-0
Degree $2$
Conductor $148$
Sign $0.721 + 0.691i$
Analytic cond. $0.0738616$
Root an. cond. $0.271774$
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  + (0.766 − 0.642i)2-s + (0.173 − 0.984i)4-s + (−1.43 + 0.524i)5-s + (−0.500 − 0.866i)8-s + (0.173 + 0.984i)9-s + (−0.766 + 1.32i)10-s + (−0.173 + 0.984i)13-s + (−0.939 − 0.342i)16-s + (−0.326 − 1.85i)17-s + (0.766 + 0.642i)18-s + (0.266 + 1.50i)20-s + (1.03 − 0.866i)25-s + (0.5 + 0.866i)26-s + (−0.173 − 0.300i)29-s + (−0.939 + 0.342i)32-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)2-s + (0.173 − 0.984i)4-s + (−1.43 + 0.524i)5-s + (−0.500 − 0.866i)8-s + (0.173 + 0.984i)9-s + (−0.766 + 1.32i)10-s + (−0.173 + 0.984i)13-s + (−0.939 − 0.342i)16-s + (−0.326 − 1.85i)17-s + (0.766 + 0.642i)18-s + (0.266 + 1.50i)20-s + (1.03 − 0.866i)25-s + (0.5 + 0.866i)26-s + (−0.173 − 0.300i)29-s + (−0.939 + 0.342i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.721 + 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.721 + 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(148\)    =    \(2^{2} \cdot 37\)
Sign: $0.721 + 0.691i$
Analytic conductor: \(0.0738616\)
Root analytic conductor: \(0.271774\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{148} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 148,\ (\ :0),\ 0.721 + 0.691i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7656358634\)
\(L(\frac12)\) \(\approx\) \(0.7656358634\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.766 + 0.642i)T \)
37 \( 1 + (-0.173 + 0.984i)T \)
good3 \( 1 + (-0.173 - 0.984i)T^{2} \)
5 \( 1 + (1.43 - 0.524i)T + (0.766 - 0.642i)T^{2} \)
7 \( 1 + (-0.766 + 0.642i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \)
17 \( 1 + (0.326 + 1.85i)T + (-0.939 + 0.342i)T^{2} \)
19 \( 1 + (-0.173 - 0.984i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
41 \( 1 + (0.326 - 1.85i)T + (-0.939 - 0.342i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \)
59 \( 1 + (-0.766 - 0.642i)T^{2} \)
61 \( 1 + (-0.0603 + 0.342i)T + (-0.939 - 0.342i)T^{2} \)
67 \( 1 + (-0.766 + 0.642i)T^{2} \)
71 \( 1 + (-0.173 - 0.984i)T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + (-0.766 + 0.642i)T^{2} \)
83 \( 1 + (0.939 - 0.342i)T^{2} \)
89 \( 1 + (-1.76 - 0.642i)T + (0.766 + 0.642i)T^{2} \)
97 \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.19207676438231685968399645970, −11.79769686394957133862561852089, −11.52041650534688665018662166600, −10.52699250978375066252027970041, −9.252173379421914501030450319945, −7.65962023900116921280015867960, −6.80232339539420751897729334144, −4.99794529017152063699270491463, −4.05554010026699310381168794944, −2.60771914128348356188111894493, 3.48199219988604857120472306775, 4.29285617711419808267934528886, 5.74595973402176295464707050931, 7.03187465897216676542790962941, 8.081708374056027259435701259860, 8.797277570315375319772675151539, 10.62476020138331775903620539922, 11.89514666250969489937966098076, 12.43857728637693839881895659731, 13.21066262419185088878117111526

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