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Properties

Label 2-152-152.107-c1-0-15
Degree $2$
Conductor $152$
Sign $0.988 - 0.152i$
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  + (1.37 + 0.334i)2-s + (1.27 − 0.736i)3-s + (1.77 + 0.918i)4-s + (−2.34 + 1.35i)5-s + (2.00 − 0.586i)6-s − 3.01i·7-s + (2.13 + 1.85i)8-s + (−0.414 + 0.717i)9-s + (−3.67 + 1.07i)10-s − 3.11·11-s + (2.94 − 0.137i)12-s + (0.218 − 0.379i)13-s + (1.00 − 4.14i)14-s + (−1.99 + 3.45i)15-s + (2.31 + 3.26i)16-s + (−3.22 − 5.58i)17-s + ⋯
L(s)  = 1  + (0.971 + 0.236i)2-s + (0.736 − 0.425i)3-s + (0.888 + 0.459i)4-s + (−1.04 + 0.605i)5-s + (0.816 − 0.239i)6-s − 1.13i·7-s + (0.754 + 0.656i)8-s + (−0.138 + 0.239i)9-s + (−1.16 + 0.340i)10-s − 0.938·11-s + (0.849 − 0.0396i)12-s + (0.0607 − 0.105i)13-s + (0.269 − 1.10i)14-s + (−0.515 + 0.892i)15-s + (0.578 + 0.815i)16-s + (−0.781 − 1.35i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.152i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 - 0.152i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $0.988 - 0.152i$
Analytic conductor: \(1.21372\)
Root analytic conductor: \(1.10169\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{152} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 152,\ (\ :1/2),\ 0.988 - 0.152i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.95368 + 0.149977i\)
\(L(\frac12)\) \(\approx\) \(1.95368 + 0.149977i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.37 - 0.334i)T \)
19 \( 1 + (-3.58 - 2.47i)T \)
good3 \( 1 + (-1.27 + 0.736i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (2.34 - 1.35i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + 3.01iT - 7T^{2} \)
11 \( 1 + 3.11T + 11T^{2} \)
13 \( 1 + (-0.218 + 0.379i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.22 + 5.58i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (1.81 + 1.04i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.58 - 2.74i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 9.77T + 31T^{2} \)
37 \( 1 + 8.36T + 37T^{2} \)
41 \( 1 + (-4.66 + 2.69i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.43 + 4.21i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.22 + 0.709i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.63 + 2.83i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.70 - 0.984i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-9.56 - 5.51i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.49 - 2.59i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.40 - 7.63i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (6.90 + 11.9i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.939 + 1.62i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.79T + 83T^{2} \)
89 \( 1 + (-10.7 - 6.21i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (16.1 - 9.29i)T + (48.5 - 84.0i)T^{2} \)
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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

−13.50624846974040517794771100402, −12.13180399185064793781416773695, −11.19536023007936956971165946912, −10.34477804472421020969837807612, −8.305540492247946109441995208990, −7.48412842921144688272232159751, −6.98492309217415775394668518431, −5.09044133263252743244051593461, −3.75208227235477264222750702953, −2.69301788785845333377924646139, 2.57664466093831577511105649868, 3.81087823289412546154567356798, 4.92829507806564699939138013008, 6.21226209535781810505749191173, 7.925498999283501252237005093246, 8.702071270794799493989799207807, 9.967631694985264924194734500018, 11.34042817318910543353075944504, 12.07405751977116212311876192639, 12.89002115135599044022859951460

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