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Properties

Label 2-153-153.50-c2-0-12
Degree $2$
Conductor $153$
Sign $0.534 - 0.844i$
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  + (−2.44 + 1.41i)2-s + (0.529 − 2.95i)3-s + (1.98 − 3.44i)4-s + (−1.84 + 3.19i)5-s + (2.87 + 7.97i)6-s + (−1.20 + 0.696i)7-s − 0.0600i·8-s + (−8.43 − 3.12i)9-s − 10.4i·10-s + (9.91 + 17.1i)11-s + (−9.12 − 7.69i)12-s + (8.79 − 15.2i)13-s + (1.96 − 3.40i)14-s + (8.45 + 7.13i)15-s + (8.04 + 13.9i)16-s + (10.4 + 13.3i)17-s + ⋯
L(s)  = 1  + (−1.22 + 0.706i)2-s + (0.176 − 0.984i)3-s + (0.497 − 0.861i)4-s + (−0.368 + 0.638i)5-s + (0.479 + 1.32i)6-s + (−0.172 + 0.0995i)7-s − 0.00750i·8-s + (−0.937 − 0.347i)9-s − 1.04i·10-s + (0.901 + 1.56i)11-s + (−0.760 − 0.641i)12-s + (0.676 − 1.17i)13-s + (0.140 − 0.243i)14-s + (0.563 + 0.475i)15-s + (0.502 + 0.870i)16-s + (0.615 + 0.788i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.534 - 0.844i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.534 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $0.534 - 0.844i$
Analytic conductor: \(4.16894\)
Root analytic conductor: \(2.04180\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{153} (50, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 153,\ (\ :1),\ 0.534 - 0.844i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.652406 + 0.359084i\)
\(L(\frac12)\) \(\approx\) \(0.652406 + 0.359084i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.529 + 2.95i)T \)
17 \( 1 + (-10.4 - 13.3i)T \)
good2 \( 1 + (2.44 - 1.41i)T + (2 - 3.46i)T^{2} \)
5 \( 1 + (1.84 - 3.19i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + (1.20 - 0.696i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (-9.91 - 17.1i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (-8.79 + 15.2i)T + (-84.5 - 146. i)T^{2} \)
19 \( 1 - 17.1T + 361T^{2} \)
23 \( 1 + (8.76 - 15.1i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-6.20 - 10.7i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-29.8 - 17.2i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 - 13.0iT - 1.36e3T^{2} \)
41 \( 1 + (-15.2 + 26.4i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (17.3 + 30.0i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-74.8 + 43.1i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 11.8iT - 2.80e3T^{2} \)
59 \( 1 + (77.9 + 45.0i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (1.04 - 0.600i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-29.0 + 50.2i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 119.T + 5.04e3T^{2} \)
73 \( 1 - 108. iT - 5.32e3T^{2} \)
79 \( 1 + (2.99 - 1.72i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (37.0 - 21.3i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 43.3iT - 7.92e3T^{2} \)
97 \( 1 + (-42.1 + 24.3i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−12.73909981526257823625411769449, −12.00003510394762576055122521601, −10.60032935517242777853985156853, −9.613030406614826862332042700872, −8.511101476706164109309180833781, −7.55210311444550414011400545678, −6.99098455346820397729691947851, −5.89983155021295811272551605052, −3.43095454411526164932107178841, −1.32348522801335950824911856750, 0.841824099194755090049308800078, 3.07417911837637451187941886857, 4.40904165812923778558585398788, 6.01914232666987106976069386570, 7.988652158221559502509470164421, 8.859846992109238861029304163368, 9.346367577786764356858230605044, 10.41684973846364466614236751521, 11.54904672375877004428647588939, 11.77687633564124219856886192325

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