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Properties

Label 2-152-19.5-c1-0-4
Degree $2$
Conductor $152$
Sign $-0.291 + 0.956i$
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  + (0.560 − 3.17i)3-s + (−1.93 − 1.62i)5-s + (1.93 + 3.35i)7-s + (−6.96 − 2.53i)9-s + (1.17 − 2.03i)11-s + (0.145 + 0.824i)13-s + (−6.25 + 5.25i)15-s + (0.900 − 0.327i)17-s + (4.35 − 0.0632i)19-s + (11.7 − 4.28i)21-s + (3.34 − 2.80i)23-s + (0.245 + 1.39i)25-s + (−7.11 + 12.3i)27-s + (3.25 + 1.18i)29-s + (2.31 + 4.01i)31-s + ⋯
L(s)  = 1  + (0.323 − 1.83i)3-s + (−0.867 − 0.727i)5-s + (0.733 + 1.26i)7-s + (−2.32 − 0.844i)9-s + (0.353 − 0.612i)11-s + (0.0403 + 0.228i)13-s + (−1.61 + 1.35i)15-s + (0.218 − 0.0794i)17-s + (0.999 − 0.0144i)19-s + (2.56 − 0.934i)21-s + (0.697 − 0.585i)23-s + (0.0490 + 0.278i)25-s + (−1.36 + 2.37i)27-s + (0.605 + 0.220i)29-s + (0.416 + 0.721i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.291 + 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.291 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $-0.291 + 0.956i$
Analytic conductor: \(1.21372\)
Root analytic conductor: \(1.10169\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{152} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 152,\ (\ :1/2),\ -0.291 + 0.956i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.667252 - 0.900691i\)
\(L(\frac12)\) \(\approx\) \(0.667252 - 0.900691i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + (-4.35 + 0.0632i)T \)
good3 \( 1 + (-0.560 + 3.17i)T + (-2.81 - 1.02i)T^{2} \)
5 \( 1 + (1.93 + 1.62i)T + (0.868 + 4.92i)T^{2} \)
7 \( 1 + (-1.93 - 3.35i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.17 + 2.03i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.145 - 0.824i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (-0.900 + 0.327i)T + (13.0 - 10.9i)T^{2} \)
23 \( 1 + (-3.34 + 2.80i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (-3.25 - 1.18i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (-2.31 - 4.01i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 6.22T + 37T^{2} \)
41 \( 1 + (1.63 - 9.25i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (3.42 + 2.87i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (-6.73 - 2.45i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (3.24 - 2.72i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (7.92 - 2.88i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (4.89 - 4.10i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-12.7 - 4.63i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (6.53 + 5.48i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (-1.54 + 8.77i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (0.647 - 3.67i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-1.23 - 2.13i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.396 + 2.24i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (-0.166 + 0.0605i)T + (74.3 - 62.3i)T^{2} \)
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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

−12.33377195608791523706800408014, −12.10972732175207549241020033326, −11.30400935490095112699782286836, −8.908652955226260928138293068524, −8.465976767063164683909598699631, −7.59583741555025952116245521914, −6.36941495451777279764713634337, −5.12223883633110212299795092473, −2.92833934409841996085641645278, −1.27680565541480310475977999137, 3.33461707007631781166032241626, 4.11487083630316234365699608302, 5.13811659221697972602746431599, 7.19023234549246917322331940599, 8.120234486119367678518246516037, 9.458998386017950943139724784716, 10.36203406261550170435953400466, 11.00748291545936435141367151961, 11.80078240223992176404943167534, 13.81020027062054880136873414123

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