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Properties

Label 2-148-148.63-c0-0-0
Degree $2$
Conductor $148$
Sign $0.367 + 0.929i$
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  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.5 − 0.866i)5-s + 0.999·8-s + (−0.5 − 0.866i)9-s − 0.999·10-s + (−1 + 1.73i)13-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.499 + 0.866i)18-s + (0.499 + 0.866i)20-s + 1.99·26-s − 29-s + (−0.499 + 0.866i)32-s + (0.499 − 0.866i)34-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.5 − 0.866i)5-s + 0.999·8-s + (−0.5 − 0.866i)9-s − 0.999·10-s + (−1 + 1.73i)13-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.499 + 0.866i)18-s + (0.499 + 0.866i)20-s + 1.99·26-s − 29-s + (−0.499 + 0.866i)32-s + (0.499 − 0.866i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.367 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.367 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(148\)    =    \(2^{2} \cdot 37\)
Sign: $0.367 + 0.929i$
Analytic conductor: \(0.0738616\)
Root analytic conductor: \(0.271774\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{148} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 148,\ (\ :0),\ 0.367 + 0.929i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5096843220\)
\(L(\frac12)\) \(\approx\) \(0.5096843220\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
good3 \( 1 + (0.5 + 0.866i)T^{2} \)
5 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + T + T^{2} \)
31 \( 1 - T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 - 2T + T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + T + T^{2} \)
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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

−12.75470100440307614342689893959, −12.10175890340696933277954386344, −11.21638905910060143805761389051, −9.742048422211040471347849514437, −9.266654306864399411552743721958, −8.287180582741234828554201407768, −6.77605692322991098383087074077, −5.11379352929463618926966154777, −3.75374885505549034519191665355, −1.86391576307011732057411614853, 2.70594643519740103407472915111, 5.07555161507183373009572892502, 5.91882122147529498271655254947, 7.32941209753467813664766886872, 7.962051909607678993465956431210, 9.426467174471067737455659293965, 10.32016329323130379036524398995, 11.03020978873252149269118182461, 12.71611033927127344535869253417, 13.85653146219689087176981720472

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