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Properties

Label 2-148-37.14-c2-0-4
Degree $2$
Conductor $148$
Sign $0.856 + 0.516i$
Analytic cond. $4.03270$
Root an. cond. $2.00816$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.99 − 2.30i)3-s + (5.23 + 1.40i)5-s + (−1.59 − 2.76i)7-s + (6.12 − 10.6i)9-s + 15.9i·11-s + (−14.8 − 3.97i)13-s + (24.1 − 6.46i)15-s + (−1.60 − 5.97i)17-s + (−9.44 − 2.53i)19-s + (−12.7 − 7.35i)21-s + (12.9 − 12.9i)23-s + (3.75 + 2.17i)25-s − 14.9i·27-s + (1.00 + 1.00i)29-s + (31.0 + 31.0i)31-s + ⋯
L(s)  = 1  + (1.33 − 0.768i)3-s + (1.04 + 0.280i)5-s + (−0.227 − 0.394i)7-s + (0.680 − 1.17i)9-s + 1.44i·11-s + (−1.14 − 0.305i)13-s + (1.60 − 0.430i)15-s + (−0.0941 − 0.351i)17-s + (−0.497 − 0.133i)19-s + (−0.606 − 0.350i)21-s + (0.563 − 0.563i)23-s + (0.150 + 0.0868i)25-s − 0.554i·27-s + (0.0346 + 0.0346i)29-s + (1.00 + 1.00i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.856 + 0.516i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 148 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.856 + 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(148\)    =    \(2^{2} \cdot 37\)
Sign: $0.856 + 0.516i$
Analytic conductor: \(4.03270\)
Root analytic conductor: \(2.00816\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{148} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 148,\ (\ :1),\ 0.856 + 0.516i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.23210 - 0.621208i\)
\(L(\frac12)\) \(\approx\) \(2.23210 - 0.621208i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
37 \( 1 + (-12.5 + 34.8i)T \)
good3 \( 1 + (-3.99 + 2.30i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (-5.23 - 1.40i)T + (21.6 + 12.5i)T^{2} \)
7 \( 1 + (1.59 + 2.76i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 - 15.9iT - 121T^{2} \)
13 \( 1 + (14.8 + 3.97i)T + (146. + 84.5i)T^{2} \)
17 \( 1 + (1.60 + 5.97i)T + (-250. + 144.5i)T^{2} \)
19 \( 1 + (9.44 + 2.53i)T + (312. + 180.5i)T^{2} \)
23 \( 1 + (-12.9 + 12.9i)T - 529iT^{2} \)
29 \( 1 + (-1.00 - 1.00i)T + 841iT^{2} \)
31 \( 1 + (-31.0 - 31.0i)T + 961iT^{2} \)
41 \( 1 + (61.1 - 35.2i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (42.8 - 42.8i)T - 1.84e3iT^{2} \)
47 \( 1 + 41.8T + 2.20e3T^{2} \)
53 \( 1 + (-35.7 + 61.9i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-17.9 - 67.0i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (9.71 - 36.2i)T + (-3.22e3 - 1.86e3i)T^{2} \)
67 \( 1 + (85.6 - 49.4i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (8.66 + 15.0i)T + (-2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 - 25.7iT - 5.32e3T^{2} \)
79 \( 1 + (-123. - 33.0i)T + (5.40e3 + 3.12e3i)T^{2} \)
83 \( 1 + (-43.6 + 75.6i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (-108. + 28.9i)T + (6.85e3 - 3.96e3i)T^{2} \)
97 \( 1 + (-20.9 + 20.9i)T - 9.40e3iT^{2} \)
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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

−13.04659619160386888721639785594, −12.10293952202744301388955219709, −10.20124194704788428044756196620, −9.709985772345662800634356110699, −8.530093246634122513734945748538, −7.30467621300814166023353538245, −6.67686318452511365291812085140, −4.79130163779199689806482658338, −2.87080624599874943416273270293, −1.90904929259969480955716436162, 2.24789237121923088084677113704, 3.43764585831076166360423838949, 5.00498922535236880320373740290, 6.28168604760097153101807678682, 8.043453786977253040071185253576, 8.944735261357774091799409846035, 9.599915466981429842204872045296, 10.47352865842382749280819847040, 11.94699980510155808028589963513, 13.46024106907711470277239146794

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