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Properties

Label 2-153-153.101-c2-0-21
Degree $2$
Conductor $153$
Sign $-0.916 + 0.401i$
Analytic cond. $4.16894$
Root an. cond. $2.04180$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.63 − 1.51i)2-s + (2.47 − 1.68i)3-s + (2.61 + 4.53i)4-s + (2.42 + 4.20i)5-s + (−9.09 + 0.681i)6-s + (−10.6 − 6.14i)7-s − 3.75i·8-s + (3.28 − 8.37i)9-s − 14.7i·10-s + (4.84 − 8.39i)11-s + (14.1 + 6.81i)12-s + (−6.92 − 11.9i)13-s + (18.6 + 32.3i)14-s + (13.1 + 6.31i)15-s + (4.76 − 8.24i)16-s + (1.46 − 16.9i)17-s + ⋯
L(s)  = 1  + (−1.31 − 0.759i)2-s + (0.826 − 0.563i)3-s + (0.654 + 1.13i)4-s + (0.485 + 0.840i)5-s + (−1.51 + 0.113i)6-s + (−1.51 − 0.877i)7-s − 0.469i·8-s + (0.365 − 0.930i)9-s − 1.47i·10-s + (0.440 − 0.762i)11-s + (1.17 + 0.568i)12-s + (−0.532 − 0.922i)13-s + (1.33 + 2.30i)14-s + (0.874 + 0.421i)15-s + (0.297 − 0.515i)16-s + (0.0858 − 0.996i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.916 + 0.401i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.916 + 0.401i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $-0.916 + 0.401i$
Analytic conductor: \(4.16894\)
Root analytic conductor: \(2.04180\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{153} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 153,\ (\ :1),\ -0.916 + 0.401i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.146014 - 0.697603i\)
\(L(\frac12)\) \(\approx\) \(0.146014 - 0.697603i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.47 + 1.68i)T \)
17 \( 1 + (-1.46 + 16.9i)T \)
good2 \( 1 + (2.63 + 1.51i)T + (2 + 3.46i)T^{2} \)
5 \( 1 + (-2.42 - 4.20i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (10.6 + 6.14i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-4.84 + 8.39i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (6.92 + 11.9i)T + (-84.5 + 146. i)T^{2} \)
19 \( 1 + 26.7T + 361T^{2} \)
23 \( 1 + (-4.23 - 7.33i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (12.5 - 21.7i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (-29.8 + 17.2i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 - 54.7iT - 1.36e3T^{2} \)
41 \( 1 + (-8.02 - 13.9i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-20.0 + 34.6i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-10.3 - 5.98i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 57.0iT - 2.80e3T^{2} \)
59 \( 1 + (0.661 - 0.381i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-56.4 - 32.5i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (52.5 + 91.0i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 72.4T + 5.04e3T^{2} \)
73 \( 1 + 20.9iT - 5.32e3T^{2} \)
79 \( 1 + (0.183 + 0.105i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-53.9 - 31.1i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 126. iT - 7.92e3T^{2} \)
97 \( 1 + (73.3 + 42.3i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−12.24488090037823011239083448550, −10.87738545920144792866099454056, −10.04358579464209559670055526674, −9.475763865663105791340537313069, −8.344912248156408272920054338425, −7.18770407048948237554602522292, −6.39125780065122306188373340676, −3.36556961059631249622026479041, −2.59069031147346392649174524707, −0.61011959233752159185748826259, 2.12305598030260540784970976298, 4.18357491314218266068272800026, 5.96866730564745523695338759688, 6.98633457005247661677858779954, 8.425553896480575732573959537566, 9.164155953437249341455429509451, 9.531478028841126479548597178788, 10.43207651381147377442568935715, 12.48090398796103981299933674496, 13.03068990651012949534871098282

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