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Properties

Label 2-148-148.119-c1-0-6
Degree $2$
Conductor $148$
Sign $0.999 + 0.00309i$
Analytic cond. $1.18178$
Root an. cond. $1.08709$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.190 − 1.40i)2-s + (−1.22 + 2.12i)3-s + (−1.92 − 0.533i)4-s + (1.77 − 0.475i)5-s + (2.74 + 2.12i)6-s + (3.91 + 2.26i)7-s + (−1.11 + 2.59i)8-s + (−1.51 − 2.63i)9-s + (−0.327 − 2.57i)10-s − 0.821·11-s + (3.50 − 3.44i)12-s + (3.40 − 0.912i)13-s + (3.91 − 5.06i)14-s + (−1.16 + 4.35i)15-s + (3.42 + 2.05i)16-s + (−0.981 + 3.66i)17-s + ⋯
L(s)  = 1  + (0.134 − 0.990i)2-s + (−0.709 + 1.22i)3-s + (−0.963 − 0.266i)4-s + (0.792 − 0.212i)5-s + (1.12 + 0.868i)6-s + (1.48 + 0.855i)7-s + (−0.394 + 0.918i)8-s + (−0.506 − 0.877i)9-s + (−0.103 − 0.814i)10-s − 0.247·11-s + (1.01 − 0.994i)12-s + (0.944 − 0.253i)13-s + (1.04 − 1.35i)14-s + (−0.301 + 1.12i)15-s + (0.857 + 0.514i)16-s + (−0.237 + 0.888i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00309i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00309i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(148\)    =    \(2^{2} \cdot 37\)
Sign: $0.999 + 0.00309i$
Analytic conductor: \(1.18178\)
Root analytic conductor: \(1.08709\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{148} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 148,\ (\ :1/2),\ 0.999 + 0.00309i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08391 - 0.00167698i\)
\(L(\frac12)\) \(\approx\) \(1.08391 - 0.00167698i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.190 + 1.40i)T \)
37 \( 1 + (-2.66 + 5.46i)T \)
good3 \( 1 + (1.22 - 2.12i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.77 + 0.475i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (-3.91 - 2.26i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + 0.821T + 11T^{2} \)
13 \( 1 + (-3.40 + 0.912i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (0.981 - 3.66i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (2.15 + 8.04i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (3.06 - 3.06i)T - 23iT^{2} \)
29 \( 1 + (-0.0398 + 0.0398i)T - 29iT^{2} \)
31 \( 1 + (-0.0904 - 0.0904i)T + 31iT^{2} \)
41 \( 1 + (5.58 + 3.22i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.33 + 4.33i)T - 43iT^{2} \)
47 \( 1 + 7.25iT - 47T^{2} \)
53 \( 1 + (3.85 + 6.67i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.22 - 0.863i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.36 - 5.09i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-2.27 + 3.94i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.33 + 1.35i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 5.66iT - 73T^{2} \)
79 \( 1 + (-0.763 - 2.85i)T + (-68.4 + 39.5i)T^{2} \)
83 \( 1 + (14.5 - 8.38i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.36 + 0.901i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (3.12 + 3.12i)T + 97iT^{2} \)
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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

−12.93414632638899705502303567885, −11.57315062078147439859792617888, −11.09364810609052261933723469737, −10.28254465961742047005182436909, −9.165837065590611030171107055042, −8.428919934092984704149882340637, −5.73982873711604174988293254047, −5.17682857955378203086469365429, −4.07981607588478913478510903022, −2.03821280699921819046152434872, 1.47518864612700292817065359304, 4.40390581683261700454879716790, 5.77110050231062680471307564630, 6.49325103643582142623613542221, 7.65108100573050001098029082317, 8.292955975638764451690374634048, 10.00888611121854740255304907417, 11.17851010950905823219125166997, 12.25037122309486805835718752991, 13.35791463549737458353374291363

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