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Properties

Label 2-148-148.103-c1-0-15
Degree $2$
Conductor $148$
Sign $0.161 + 0.986i$
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  + (0.946 − 1.05i)2-s + (0.702 − 1.21i)3-s + (−0.208 − 1.98i)4-s + (0.462 + 1.72i)5-s + (−0.614 − 1.89i)6-s + (−0.779 − 0.450i)7-s + (−2.28 − 1.66i)8-s + (0.512 + 0.887i)9-s + (2.25 + 1.14i)10-s − 1.34·11-s + (−2.56 − 1.14i)12-s + (0.709 + 2.64i)13-s + (−1.21 + 0.393i)14-s + (2.42 + 0.649i)15-s + (−3.91 + 0.830i)16-s + (−2.80 − 0.752i)17-s + ⋯
L(s)  = 1  + (0.669 − 0.743i)2-s + (0.405 − 0.702i)3-s + (−0.104 − 0.994i)4-s + (0.206 + 0.771i)5-s + (−0.250 − 0.771i)6-s + (−0.294 − 0.170i)7-s + (−0.808 − 0.587i)8-s + (0.170 + 0.295i)9-s + (0.711 + 0.362i)10-s − 0.406·11-s + (−0.741 − 0.330i)12-s + (0.196 + 0.734i)13-s + (−0.323 + 0.105i)14-s + (0.625 + 0.167i)15-s + (−0.978 + 0.207i)16-s + (−0.681 − 0.182i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.161 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.161 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(148\)    =    \(2^{2} \cdot 37\)
Sign: $0.161 + 0.986i$
Analytic conductor: \(1.18178\)
Root analytic conductor: \(1.08709\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{148} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 148,\ (\ :1/2),\ 0.161 + 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25679 - 1.06839i\)
\(L(\frac12)\) \(\approx\) \(1.25679 - 1.06839i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.946 + 1.05i)T \)
37 \( 1 + (-5.34 - 2.90i)T \)
good3 \( 1 + (-0.702 + 1.21i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.462 - 1.72i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (0.779 + 0.450i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + 1.34T + 11T^{2} \)
13 \( 1 + (-0.709 - 2.64i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (2.80 + 0.752i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-0.770 + 0.206i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-1.26 - 1.26i)T + 23iT^{2} \)
29 \( 1 + (-0.0924 - 0.0924i)T + 29iT^{2} \)
31 \( 1 + (2.26 - 2.26i)T - 31iT^{2} \)
41 \( 1 + (5.96 + 3.44i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (8.81 + 8.81i)T + 43iT^{2} \)
47 \( 1 - 9.02iT - 47T^{2} \)
53 \( 1 + (2.28 + 3.96i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.43 - 12.8i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-8.81 + 2.36i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (-4.19 + 7.26i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (13.2 + 7.66i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 13.9iT - 73T^{2} \)
79 \( 1 + (7.92 - 2.12i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (-5.47 + 3.16i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.325 - 1.21i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (1.92 - 1.92i)T - 97iT^{2} \)
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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

−13.07396013593838713307705464803, −11.90498003532070212422508507013, −10.88031806167551660870639363258, −10.07544479784211642692982376013, −8.799461564660624610278582546171, −7.20691265391480404899499823951, −6.39954151499023330927266230924, −4.80767049216300370021764301736, −3.19675707425274738730149932548, −1.97146832917010157856392705964, 3.09894616048121474247390150745, 4.39385499088317745012481401932, 5.42350845170632714646697091504, 6.68534947159114011713766785951, 8.153465461430604444067379309407, 8.964161315069792939790850760612, 9.956045449873822493893498761801, 11.43755074917639281113132609049, 12.90116769337717684908376191621, 13.00738510455290873019757812388

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