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Properties

Label 2-153-153.101-c2-0-31
Degree $2$
Conductor $153$
Sign $-0.374 + 0.927i$
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  + (1.23 + 0.713i)2-s + (−0.621 + 2.93i)3-s + (−0.982 − 1.70i)4-s + (−4.13 − 7.15i)5-s + (−2.86 + 3.18i)6-s + (−6.73 − 3.89i)7-s − 8.50i·8-s + (−8.22 − 3.64i)9-s − 11.7i·10-s + (−8.39 + 14.5i)11-s + (5.60 − 1.82i)12-s + (4.49 + 7.78i)13-s + (−5.55 − 9.61i)14-s + (23.5 − 7.67i)15-s + (2.13 − 3.70i)16-s + (12.6 − 11.3i)17-s + ⋯
L(s)  = 1  + (0.617 + 0.356i)2-s + (−0.207 + 0.978i)3-s + (−0.245 − 0.425i)4-s + (−0.826 − 1.43i)5-s + (−0.476 + 0.530i)6-s + (−0.962 − 0.555i)7-s − 1.06i·8-s + (−0.914 − 0.405i)9-s − 1.17i·10-s + (−0.763 + 1.32i)11-s + (0.467 − 0.152i)12-s + (0.345 + 0.598i)13-s + (−0.396 − 0.686i)14-s + (1.57 − 0.511i)15-s + (0.133 − 0.231i)16-s + (0.743 − 0.668i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.374 + 0.927i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $-0.374 + 0.927i$
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: \((2,\ 153,\ (\ :1),\ -0.374 + 0.927i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.360513 - 0.534341i\)
\(L(\frac12)\) \(\approx\) \(0.360513 - 0.534341i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.621 - 2.93i)T \)
17 \( 1 + (-12.6 + 11.3i)T \)
good2 \( 1 + (-1.23 - 0.713i)T + (2 + 3.46i)T^{2} \)
5 \( 1 + (4.13 + 7.15i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (6.73 + 3.89i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (8.39 - 14.5i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (-4.49 - 7.78i)T + (-84.5 + 146. i)T^{2} \)
19 \( 1 + 1.95T + 361T^{2} \)
23 \( 1 + (1.73 + 3.00i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-27.4 + 47.5i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (11.6 - 6.75i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + 26.1iT - 1.36e3T^{2} \)
41 \( 1 + (12.7 + 22.1i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (20.3 - 35.2i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (61.0 + 35.2i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 90.0iT - 2.80e3T^{2} \)
59 \( 1 + (-40.2 + 23.2i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-28.8 - 16.6i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-35.7 - 61.9i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 33.4T + 5.04e3T^{2} \)
73 \( 1 - 83.7iT - 5.32e3T^{2} \)
79 \( 1 + (63.6 + 36.7i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (65.6 + 37.8i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 90.7iT - 7.92e3T^{2} \)
97 \( 1 + (42.2 + 24.4i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−12.61317935765343763622914975508, −11.62373925077678406290433290368, −9.955208859794010046171210951160, −9.712570187743454736584837740399, −8.370376853366240309615534809561, −6.87045695530882956360495877674, −5.38907315861437552532346236433, −4.58574071832705070883447795347, −3.80507543628086358260844517818, −0.33857166346297578063180204403, 3.00446963508907490384591966269, 3.25353934335530350700259047132, 5.57798663396213600083562476709, 6.55638102337528092381510626495, 7.81780902348177578037984709169, 8.501612271685349100768950844170, 10.54922569219688289935476977563, 11.26055886031989956829608710999, 12.21547722539728940457176147013, 12.92354684260934442901810624737

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