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Properties

Label 2-148-148.119-c1-0-4
Degree $2$
Conductor $148$
Sign $-0.744 - 0.667i$
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  + (0.795 + 1.16i)2-s + (−0.610 + 1.05i)3-s + (−0.734 + 1.86i)4-s + (−2.16 + 0.580i)5-s + (−1.72 + 0.127i)6-s + (−1.35 − 0.783i)7-s + (−2.75 + 0.621i)8-s + (0.754 + 1.30i)9-s + (−2.40 − 2.07i)10-s + 5.47·11-s + (−1.51 − 1.91i)12-s + (3.96 − 1.06i)13-s + (−0.163 − 2.21i)14-s + (0.709 − 2.64i)15-s + (−2.92 − 2.73i)16-s + (−1.66 + 6.20i)17-s + ⋯
L(s)  = 1  + (0.562 + 0.826i)2-s + (−0.352 + 0.610i)3-s + (−0.367 + 0.930i)4-s + (−0.969 + 0.259i)5-s + (−0.703 + 0.0519i)6-s + (−0.512 − 0.296i)7-s + (−0.975 + 0.219i)8-s + (0.251 + 0.435i)9-s + (−0.760 − 0.655i)10-s + 1.65·11-s + (−0.438 − 0.552i)12-s + (1.09 − 0.294i)13-s + (−0.0436 − 0.590i)14-s + (0.183 − 0.683i)15-s + (−0.730 − 0.683i)16-s + (−0.403 + 1.50i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.744 - 0.667i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.744 - 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(148\)    =    \(2^{2} \cdot 37\)
Sign: $-0.744 - 0.667i$
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: \((2,\ 148,\ (\ :1/2),\ -0.744 - 0.667i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.391152 + 1.02131i\)
\(L(\frac12)\) \(\approx\) \(0.391152 + 1.02131i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.795 - 1.16i)T \)
37 \( 1 + (2.88 + 5.35i)T \)
good3 \( 1 + (0.610 - 1.05i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (2.16 - 0.580i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (1.35 + 0.783i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 - 5.47T + 11T^{2} \)
13 \( 1 + (-3.96 + 1.06i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (1.66 - 6.20i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (0.0263 + 0.0981i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-1.51 + 1.51i)T - 23iT^{2} \)
29 \( 1 + (-4.58 + 4.58i)T - 29iT^{2} \)
31 \( 1 + (-0.192 - 0.192i)T + 31iT^{2} \)
41 \( 1 + (-1.34 - 0.774i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-8.79 + 8.79i)T - 43iT^{2} \)
47 \( 1 - 6.18iT - 47T^{2} \)
53 \( 1 + (-0.340 - 0.589i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.27 - 0.610i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (0.732 + 2.73i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (4.01 - 6.95i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.43 - 1.40i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 6.47iT - 73T^{2} \)
79 \( 1 + (-0.298 - 1.11i)T + (-68.4 + 39.5i)T^{2} \)
83 \( 1 + (11.8 - 6.83i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-11.6 - 3.11i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-0.805 - 0.805i)T + 97iT^{2} \)
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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

−13.53568943681614505789085047695, −12.51968713146709185168049997930, −11.49018296777091816786526296372, −10.57693150354217091814879042592, −9.093192300607405147786325412772, −8.062957091017438979483681464906, −6.84566295832986384938157824641, −5.92818942612782058538795023145, −4.15857578621103029752936612534, −3.79542471825830751968991471272, 1.10336432820525434650308106704, 3.38951489477051361178690386425, 4.44366436599610200635640093835, 6.15739357807324338442176497660, 6.95951917008102405403246925583, 8.813044829132870633807966347503, 9.546073686746408714587669249039, 11.22017570414159092106777506705, 11.80171389190819743769790846983, 12.37359789963278644637881953967

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