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Properties

Label 2-148-148.7-c0-0-0
Degree $2$
Conductor $148$
Sign $0.918 + 0.395i$
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  + (−0.939 + 0.342i)2-s + (0.766 − 0.642i)4-s + (−0.326 − 1.85i)5-s + (−0.500 + 0.866i)8-s + (0.766 + 0.642i)9-s + (0.939 + 1.62i)10-s + (−0.766 + 0.642i)13-s + (0.173 − 0.984i)16-s + (0.266 + 0.223i)17-s + (−0.939 − 0.342i)18-s + (−1.43 − 1.20i)20-s + (−2.37 + 0.866i)25-s + (0.5 − 0.866i)26-s + (−0.766 + 1.32i)29-s + (0.173 + 0.984i)32-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)2-s + (0.766 − 0.642i)4-s + (−0.326 − 1.85i)5-s + (−0.500 + 0.866i)8-s + (0.766 + 0.642i)9-s + (0.939 + 1.62i)10-s + (−0.766 + 0.642i)13-s + (0.173 − 0.984i)16-s + (0.266 + 0.223i)17-s + (−0.939 − 0.342i)18-s + (−1.43 − 1.20i)20-s + (−2.37 + 0.866i)25-s + (0.5 − 0.866i)26-s + (−0.766 + 1.32i)29-s + (0.173 + 0.984i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 + 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 + 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(148\)    =    \(2^{2} \cdot 37\)
Sign: $0.918 + 0.395i$
Analytic conductor: \(0.0738616\)
Root analytic conductor: \(0.271774\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{148} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 148,\ (\ :0),\ 0.918 + 0.395i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4329891333\)
\(L(\frac12)\) \(\approx\) \(0.4329891333\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.939 - 0.342i)T \)
37 \( 1 + (-0.766 + 0.642i)T \)
good3 \( 1 + (-0.766 - 0.642i)T^{2} \)
5 \( 1 + (0.326 + 1.85i)T + (-0.939 + 0.342i)T^{2} \)
7 \( 1 + (0.939 - 0.342i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \)
17 \( 1 + (-0.266 - 0.223i)T + (0.173 + 0.984i)T^{2} \)
19 \( 1 + (-0.766 - 0.642i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
41 \( 1 + (-0.266 + 0.223i)T + (0.173 - 0.984i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \)
59 \( 1 + (0.939 + 0.342i)T^{2} \)
61 \( 1 + (-1.17 + 0.984i)T + (0.173 - 0.984i)T^{2} \)
67 \( 1 + (0.939 - 0.342i)T^{2} \)
71 \( 1 + (-0.766 - 0.642i)T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + (0.939 - 0.342i)T^{2} \)
83 \( 1 + (-0.173 - 0.984i)T^{2} \)
89 \( 1 + (-0.0603 + 0.342i)T + (-0.939 - 0.342i)T^{2} \)
97 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
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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

−12.96985084603377907476145941429, −12.26058285042900701455562586426, −11.12682541106870582177165985682, −9.799001491035983925757617034976, −9.083746026979643095993861106402, −8.091701169976091007382258041115, −7.22126222370582224323610829624, −5.50316130092364474241279526376, −4.49932666237593842736732035913, −1.63750977340220139756058364922, 2.58179836465354088901631358036, 3.73765694704081880445300521503, 6.29270531474065303935794665906, 7.19586654304967017284530179285, 7.921398222938903653264977496491, 9.697213767175371508882180481746, 10.15107305307760759056592893071, 11.22187416142163031799349253842, 11.95136687191766605023405784006, 13.17187479670503086116915745229

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