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Properties

Label 2-153-17.15-c1-0-6
Degree $2$
Conductor $153$
Sign $-0.998 - 0.0465i$
Analytic cond. $1.22171$
Root an. cond. $1.10531$
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 − 0.707i)2-s − 0.999i·4-s + (−3.12 + 1.29i)5-s + (−3.41 − 1.41i)7-s + (−2.12 + 2.12i)8-s + (3.12 + 1.29i)10-s + (2 − 4.82i)11-s + 4.24i·13-s + (1.41 + 3.41i)14-s + 1.00·16-s + (−3 − 2.82i)17-s + (−2 − 2i)19-s + (1.29 + 3.12i)20-s + (−4.82 + 1.99i)22-s + (0.242 − 0.585i)23-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s − 0.499i·4-s + (−1.39 + 0.578i)5-s + (−1.29 − 0.534i)7-s + (−0.750 + 0.750i)8-s + (0.987 + 0.408i)10-s + (0.603 − 1.45i)11-s + 1.17i·13-s + (0.377 + 0.912i)14-s + 0.250·16-s + (−0.727 − 0.685i)17-s + (−0.458 − 0.458i)19-s + (0.289 + 0.697i)20-s + (−1.02 + 0.426i)22-s + (0.0505 − 0.122i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0465i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0465i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $-0.998 - 0.0465i$
Analytic conductor: \(1.22171\)
Root analytic conductor: \(1.10531\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{153} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 153,\ (\ :1/2),\ -0.998 - 0.0465i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00615451 + 0.264490i\)
\(L(\frac12)\) \(\approx\) \(0.00615451 + 0.264490i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 + (3 + 2.82i)T \)
good2 \( 1 + (0.707 + 0.707i)T + 2iT^{2} \)
5 \( 1 + (3.12 - 1.29i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (3.41 + 1.41i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (-2 + 4.82i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 - 4.24iT - 13T^{2} \)
19 \( 1 + (2 + 2i)T + 19iT^{2} \)
23 \( 1 + (-0.242 + 0.585i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (-4.12 + 1.70i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + (0.585 + 1.41i)T + (-21.9 + 21.9i)T^{2} \)
37 \( 1 + (0.949 + 2.29i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (4.12 + 1.70i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (-4 + 4i)T - 43iT^{2} \)
47 \( 1 - 2.82iT - 47T^{2} \)
53 \( 1 + (-0.171 - 0.171i)T + 53iT^{2} \)
59 \( 1 + (3.17 - 3.17i)T - 59iT^{2} \)
61 \( 1 + (11.5 + 4.77i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 - 4.48T + 67T^{2} \)
71 \( 1 + (0.242 + 0.585i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (8.53 - 3.53i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (3.07 - 7.41i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (8.82 + 8.82i)T + 83iT^{2} \)
89 \( 1 - 10.5iT - 89T^{2} \)
97 \( 1 + (-8.94 + 3.70i)T + (68.5 - 68.5i)T^{2} \)
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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

−12.03619601362614244926215051753, −11.29799500862676691433420594277, −10.66886982294961104509820568247, −9.393706291016180848516510970692, −8.608119803077201843988370886949, −7.01961783478029715834157480514, −6.27034520207657539988280631038, −4.17620480802856012854646608749, −2.99425463296512682010279144201, −0.29228803912476025806002370331, 3.26187294061099393285011693578, 4.39643020883773294382262801234, 6.33141493857099960021940037830, 7.32696802101076388397711225054, 8.292694143670151206448750447585, 9.102132262607757806679009351547, 10.20560669854388477309488665205, 11.86406107836548825271819553489, 12.55665491727654455822428684816, 12.90476524247373321346491304655

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