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Properties

Label 2-152-152.107-c1-0-6
Degree $2$
Conductor $152$
Sign $0.977 + 0.211i$
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  + (−0.707 − 1.22i)2-s + (−1.5 + 0.866i)3-s + (−0.999 + 1.73i)4-s + (2.12 − 1.22i)5-s + (2.12 + 1.22i)6-s + 2.44i·7-s + 2.82·8-s + (−3 − 1.73i)10-s + 5·11-s − 3.46i·12-s + (2.82 − 4.89i)13-s + (2.99 − 1.73i)14-s + (−2.12 + 3.67i)15-s + (−2.00 − 3.46i)16-s + (2 + 3.46i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.866i)2-s + (−0.866 + 0.499i)3-s + (−0.499 + 0.866i)4-s + (0.948 − 0.547i)5-s + (0.866 + 0.499i)6-s + 0.925i·7-s + 0.999·8-s + (−0.948 − 0.547i)10-s + 1.50·11-s − 0.999i·12-s + (0.784 − 1.35i)13-s + (0.801 − 0.462i)14-s + (−0.547 + 0.948i)15-s + (−0.500 − 0.866i)16-s + (0.485 + 0.840i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.211i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $0.977 + 0.211i$
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: \((2,\ 152,\ (\ :1/2),\ 0.977 + 0.211i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.789569 - 0.0843498i\)
\(L(\frac12)\) \(\approx\) \(0.789569 - 0.0843498i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 1.22i)T \)
19 \( 1 + (0.5 - 4.33i)T \)
good3 \( 1 + (1.5 - 0.866i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-2.12 + 1.22i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 - 2.44iT - 7T^{2} \)
11 \( 1 - 5T + 11T^{2} \)
13 \( 1 + (-2.82 + 4.89i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2 - 3.46i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (2.12 + 1.22i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.53 - 6.12i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.41T + 31T^{2} \)
37 \( 1 + 1.41T + 37T^{2} \)
41 \( 1 + (-4.5 + 2.59i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (3 + 5.19i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.36 + 3.67i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.24 + 7.34i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.5 - 0.866i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.36 + 3.67i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.5 + 6.06i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.41 - 2.44i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.5 + 2.59i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.41 + 2.44i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 7T + 83T^{2} \)
89 \( 1 + (-3 - 1.73i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (16.5 - 9.52i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.54363987920351350149008710104, −11.95811437252673232917927766836, −10.81298387769657530973896670361, −10.07309923427370087062094524665, −9.069026717851080374294146002857, −8.235057198727222042747447229985, −6.07384442338997915710984188641, −5.31235737585186137063396209782, −3.67967678385558489364021813482, −1.64170490887695556239956411996, 1.30211196639375072025875228599, 4.29964433841244378622899928889, 6.01056715488605430613554172176, 6.52591937501407934536063178829, 7.29817369690709708420333296027, 9.076381649939106874400794488403, 9.745490755354107276481820422760, 11.03613521286630494273113413799, 11.73960358717213504714736259184, 13.50924193946596060383041804656

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