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Properties

Label 2-152-152.3-c1-0-7
Degree $2$
Conductor $152$
Sign $0.973 + 0.228i$
Analytic cond. $1.21372$
Root an. cond. $1.10169$
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  + (−0.909 + 1.08i)2-s + (−0.301 − 0.827i)3-s + (−0.347 − 1.96i)4-s + (1.17 + 0.426i)6-s + (2.44 + 1.41i)8-s + (1.70 − 1.42i)9-s + (3.30 − 5.72i)11-s + (−1.52 + 0.881i)12-s + (−3.75 + 1.36i)16-s + (6.05 + 5.07i)17-s + 3.14i·18-s + (−4.35 − 0.248i)19-s + (3.19 + 8.77i)22-s + (0.432 − 2.45i)24-s + (−4.69 − 1.71i)25-s + ⋯
L(s)  = 1  + (−0.642 + 0.766i)2-s + (−0.173 − 0.478i)3-s + (−0.173 − 0.984i)4-s + (0.478 + 0.173i)6-s + (0.866 + 0.500i)8-s + (0.567 − 0.476i)9-s + (0.995 − 1.72i)11-s + (−0.440 + 0.254i)12-s + (−0.939 + 0.342i)16-s + (1.46 + 1.23i)17-s + 0.741i·18-s + (−0.998 − 0.0569i)19-s + (0.681 + 1.87i)22-s + (0.0883 − 0.500i)24-s + (−0.939 − 0.342i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.228i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 + 0.228i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $0.973 + 0.228i$
Analytic conductor: \(1.21372\)
Root analytic conductor: \(1.10169\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{152} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 152,\ (\ :1/2),\ 0.973 + 0.228i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.813199 - 0.0942721i\)
\(L(\frac12)\) \(\approx\) \(0.813199 - 0.0942721i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.909 - 1.08i)T \)
19 \( 1 + (4.35 + 0.248i)T \)
good3 \( 1 + (0.301 + 0.827i)T + (-2.29 + 1.92i)T^{2} \)
5 \( 1 + (4.69 + 1.71i)T^{2} \)
7 \( 1 + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3.30 + 5.72i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-6.05 - 5.07i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (21.6 - 7.86i)T^{2} \)
29 \( 1 + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 + (-2.69 - 7.39i)T + (-31.4 + 26.3i)T^{2} \)
43 \( 1 + (-0.407 + 2.31i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-8.16 + 46.2i)T^{2} \)
53 \( 1 + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (7.22 - 8.60i)T + (-10.2 - 58.1i)T^{2} \)
61 \( 1 + (57.3 - 20.8i)T^{2} \)
67 \( 1 + (-9.80 - 11.6i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (-3.68 + 1.34i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-2.95 - 5.11i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-6.29 + 17.3i)T + (-68.1 - 57.2i)T^{2} \)
97 \( 1 + (4.43 - 5.28i)T + (-16.8 - 95.5i)T^{2} \)
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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

−13.07482231180679303543512066997, −11.89594468628853079833119689944, −10.80078428672623333388142665165, −9.734985002909662170362569711242, −8.623048456520989462616240695639, −7.77569394510154983700580920265, −6.38399005978254315131365728324, −5.90973811349493966543244836810, −3.92052484443969237383655736151, −1.21541994548878993584620893262, 1.90200545587019748011711658207, 3.83484545583305213281099402193, 4.89751354304518334996476133390, 6.98003123486786605352473131954, 7.85843426211155480476401217387, 9.487318723284728690832799457103, 9.788292758988419294752434454636, 10.88888174547306215194455936554, 12.00298418444139303176535939512, 12.60813916181447631343762511840

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