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Properties

Label 2-152-152.107-c1-0-13
Degree $2$
Conductor $152$
Sign $0.924 + 0.380i$
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  + (−1.22 + 0.707i)2-s + (2.72 − 1.57i)3-s + (0.999 − 1.73i)4-s + (−2.22 + 3.85i)6-s + 2.82i·8-s + (3.44 − 5.97i)9-s − 5.44·11-s − 6.29i·12-s + (−2.00 − 3.46i)16-s + (3 + 5.19i)17-s + 9.75i·18-s + (4.17 + 1.25i)19-s + (6.67 − 3.85i)22-s + (4.44 + 7.70i)24-s + (−2.5 + 4.33i)25-s + ⋯
L(s)  = 1  + (−0.866 + 0.499i)2-s + (1.57 − 0.908i)3-s + (0.499 − 0.866i)4-s + (−0.908 + 1.57i)6-s + 0.999i·8-s + (1.14 − 1.99i)9-s − 1.64·11-s − 1.81i·12-s + (−0.500 − 0.866i)16-s + (0.727 + 1.26i)17-s + 2.29i·18-s + (0.957 + 0.287i)19-s + (1.42 − 0.821i)22-s + (0.908 + 1.57i)24-s + (−0.5 + 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.924 + 0.380i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.924 + 0.380i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $0.924 + 0.380i$
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: \((2,\ 152,\ (\ :1/2),\ 0.924 + 0.380i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14848 - 0.226845i\)
\(L(\frac12)\) \(\approx\) \(1.14848 - 0.226845i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 - 0.707i)T \)
19 \( 1 + (-4.17 - 1.25i)T \)
good3 \( 1 + (-2.72 + 1.57i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (2.5 - 4.33i)T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 + 5.44T + 11T^{2} \)
13 \( 1 + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 + (9.39 - 5.42i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (5 + 8.66i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.62 - 0.937i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (14.1 + 8.18i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-6.84 - 11.8i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 11.4T + 83T^{2} \)
89 \( 1 + (4.89 + 2.82i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.151 + 0.0874i)T + (48.5 - 84.0i)T^{2} \)
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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

−13.24619562870985630114810328047, −12.10164365384095695643156366228, −10.46959907772727434618875629433, −9.609561455739906533788392868993, −8.424270429317701597663374835945, −7.890867231086314302489125744458, −7.08655341815024528902979291113, −5.60020165533225881007189454716, −3.17108512385341525215139495822, −1.75230838598433012350607007748, 2.49116991143805464815587785016, 3.33768690329194009857447631291, 4.93325195051411985730614126702, 7.45011833342186275059922096136, 8.043153497958417144149922295756, 9.076523297187303500062741768657, 9.938033770041383107275568835680, 10.50115548253728582991598299705, 11.87586107985715897784247793769, 13.26318121923660802182341463589

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