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Properties

Label 2-153-153.101-c2-0-3
Degree $2$
Conductor $153$
Sign $-0.252 - 0.967i$
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  + (−0.804 − 0.464i)2-s + (−0.168 + 2.99i)3-s + (−1.56 − 2.71i)4-s + (0.197 + 0.342i)5-s + (1.52 − 2.33i)6-s + (4.55 + 2.63i)7-s + 6.63i·8-s + (−8.94 − 1.01i)9-s − 0.367i·10-s + (−4.41 + 7.65i)11-s + (8.40 − 4.23i)12-s + (6.14 + 10.6i)13-s + (−2.44 − 4.23i)14-s + (−1.05 + 0.534i)15-s + (−3.19 + 5.52i)16-s + (−8.22 + 14.8i)17-s + ⋯
L(s)  = 1  + (−0.402 − 0.232i)2-s + (−0.0562 + 0.998i)3-s + (−0.392 − 0.679i)4-s + (0.0395 + 0.0685i)5-s + (0.254 − 0.388i)6-s + (0.651 + 0.376i)7-s + 0.828i·8-s + (−0.993 − 0.112i)9-s − 0.0367i·10-s + (−0.401 + 0.695i)11-s + (0.700 − 0.353i)12-s + (0.472 + 0.818i)13-s + (−0.174 − 0.302i)14-s + (−0.0706 + 0.0356i)15-s + (−0.199 + 0.345i)16-s + (−0.483 + 0.875i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.252 - 0.967i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $-0.252 - 0.967i$
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: \((2,\ 153,\ (\ :1),\ -0.252 - 0.967i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.497819 + 0.644476i\)
\(L(\frac12)\) \(\approx\) \(0.497819 + 0.644476i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.168 - 2.99i)T \)
17 \( 1 + (8.22 - 14.8i)T \)
good2 \( 1 + (0.804 + 0.464i)T + (2 + 3.46i)T^{2} \)
5 \( 1 + (-0.197 - 0.342i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (-4.55 - 2.63i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (4.41 - 7.65i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (-6.14 - 10.6i)T + (-84.5 + 146. i)T^{2} \)
19 \( 1 + 35.2T + 361T^{2} \)
23 \( 1 + (-17.7 - 30.7i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (23.3 - 40.4i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (-49.8 + 28.7i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 - 30.4iT - 1.36e3T^{2} \)
41 \( 1 + (10.8 + 18.8i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-16.7 + 29.0i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-9.35 - 5.40i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 17.7iT - 2.80e3T^{2} \)
59 \( 1 + (-54.8 + 31.6i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (27.3 + 15.8i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (10.6 + 18.3i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 0.0974T + 5.04e3T^{2} \)
73 \( 1 - 92.5iT - 5.32e3T^{2} \)
79 \( 1 + (-8.59 - 4.96i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (134. + 77.3i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 74.8iT - 7.92e3T^{2} \)
97 \( 1 + (-145. - 84.2i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−13.08711160589100716146174288643, −11.59042351443147196264257820217, −10.81534374523448408336508879263, −10.09117360243491419223362244467, −8.964754856296148564034840154803, −8.390879645816844332160111314433, −6.37434568729752379001989215116, −5.12658605649415136941663182703, −4.22185970874484787145637929296, −2.04443864946409154691532529800, 0.60248909496583725947031278758, 2.80957995931913777671035854738, 4.60224934194946706775254040097, 6.22131805368852582205227732698, 7.34381787018759183350205687240, 8.290737038191429907042938757893, 8.811691261200963556461706532204, 10.58469937149423611292301769067, 11.46317176017706382620127526501, 12.77388942999814908905461313110

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