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Properties

Label 2-153-17.16-c9-0-56
Degree $2$
Conductor $153$
Sign $-0.894 + 0.448i$
Analytic cond. $78.8004$
Root an. cond. $8.87696$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9.15·2-s − 428.·4-s − 1.75e3i·5-s − 5.86e3i·7-s − 8.60e3·8-s − 1.60e4i·10-s − 5.72e4i·11-s + 1.83e5·13-s − 5.37e4i·14-s + 1.40e5·16-s + (3.07e5 − 1.54e5i)17-s + 2.69e5·19-s + 7.50e5i·20-s − 5.23e5i·22-s − 1.82e6i·23-s + ⋯
L(s)  = 1  + 0.404·2-s − 0.836·4-s − 1.25i·5-s − 0.924i·7-s − 0.742·8-s − 0.507i·10-s − 1.17i·11-s + 1.77·13-s − 0.373i·14-s + 0.535·16-s + (0.894 − 0.448i)17-s + 0.474·19-s + 1.04i·20-s − 0.476i·22-s − 1.36i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.448i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $-0.894 + 0.448i$
Analytic conductor: \(78.8004\)
Root analytic conductor: \(8.87696\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{153} (118, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 153,\ (\ :9/2),\ -0.894 + 0.448i)\)

Particular Values

\(L(5)\) \(\approx\) \(2.263851084\)
\(L(\frac12)\) \(\approx\) \(2.263851084\)
\(L(\frac{11}{2})\) 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 \( 1 + (-3.07e5 + 1.54e5i)T \)
good2 \( 1 - 9.15T + 512T^{2} \)
5 \( 1 + 1.75e3iT - 1.95e6T^{2} \)
7 \( 1 + 5.86e3iT - 4.03e7T^{2} \)
11 \( 1 + 5.72e4iT - 2.35e9T^{2} \)
13 \( 1 - 1.83e5T + 1.06e10T^{2} \)
19 \( 1 - 2.69e5T + 3.22e11T^{2} \)
23 \( 1 + 1.82e6iT - 1.80e12T^{2} \)
29 \( 1 + 1.33e6iT - 1.45e13T^{2} \)
31 \( 1 + 5.82e6iT - 2.64e13T^{2} \)
37 \( 1 - 1.79e7iT - 1.29e14T^{2} \)
41 \( 1 + 1.95e7iT - 3.27e14T^{2} \)
43 \( 1 - 1.23e7T + 5.02e14T^{2} \)
47 \( 1 + 3.64e7T + 1.11e15T^{2} \)
53 \( 1 - 4.39e7T + 3.29e15T^{2} \)
59 \( 1 + 6.25e7T + 8.66e15T^{2} \)
61 \( 1 + 1.82e8iT - 1.16e16T^{2} \)
67 \( 1 - 1.37e8T + 2.72e16T^{2} \)
71 \( 1 - 5.84e7iT - 4.58e16T^{2} \)
73 \( 1 - 1.40e8iT - 5.88e16T^{2} \)
79 \( 1 - 1.26e8iT - 1.19e17T^{2} \)
83 \( 1 + 9.17e7T + 1.86e17T^{2} \)
89 \( 1 - 5.55e8T + 3.50e17T^{2} \)
97 \( 1 - 6.97e8iT - 7.60e17T^{2} \)
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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

−10.89438249488222537790123813071, −9.672715654085552283366468924615, −8.626659763857933606861320517343, −8.106336671139937078532608968366, −6.23693492711388795142964818348, −5.27579965822648490530921017724, −4.20304268493332412274549764449, −3.38105566371735159451338924006, −0.965462868633772640525243754990, −0.64570807652862879708778378440, 1.45465917287925197928624512529, 3.02138572656329586917481763616, 3.83255241477931829108142414376, 5.35333443522754270848004237343, 6.18332045486781488259702593396, 7.47767994718616303033221178057, 8.713914440508890451021855374366, 9.661100072236200127227949752502, 10.69076283785322589440330948699, 11.80056387623517360034850557997

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