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Properties

Label 2-153-153.106-c1-0-8
Degree $2$
Conductor $153$
Sign $0.456 - 0.889i$
Analytic cond. $1.22171$
Root an. cond. $1.10531$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.64 + 0.949i)2-s + (1.34 + 1.09i)3-s + (0.801 + 1.38i)4-s + (−2.17 − 0.583i)5-s + (1.16 + 3.07i)6-s + (−1.92 + 0.516i)7-s − 0.752i·8-s + (0.598 + 2.93i)9-s + (−3.02 − 3.02i)10-s + (5.69 − 1.52i)11-s + (−0.446 + 2.74i)12-s + (−2.62 − 4.53i)13-s + (−3.65 − 0.979i)14-s + (−2.28 − 3.17i)15-s + (2.31 − 4.01i)16-s + (−2.46 + 3.30i)17-s + ⋯
L(s)  = 1  + (1.16 + 0.671i)2-s + (0.774 + 0.632i)3-s + (0.400 + 0.694i)4-s + (−0.974 − 0.261i)5-s + (0.475 + 1.25i)6-s + (−0.728 + 0.195i)7-s − 0.266i·8-s + (0.199 + 0.979i)9-s + (−0.957 − 0.957i)10-s + (1.71 − 0.459i)11-s + (−0.128 + 0.791i)12-s + (−0.726 − 1.25i)13-s + (−0.977 − 0.261i)14-s + (−0.589 − 0.818i)15-s + (0.579 − 1.00i)16-s + (−0.596 + 0.802i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.456 - 0.889i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.456 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $0.456 - 0.889i$
Analytic conductor: \(1.22171\)
Root analytic conductor: \(1.10531\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{153} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 153,\ (\ :1/2),\ 0.456 - 0.889i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.72068 + 1.05140i\)
\(L(\frac12)\) \(\approx\) \(1.72068 + 1.05140i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.34 - 1.09i)T \)
17 \( 1 + (2.46 - 3.30i)T \)
good2 \( 1 + (-1.64 - 0.949i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (2.17 + 0.583i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (1.92 - 0.516i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-5.69 + 1.52i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (2.62 + 4.53i)T + (-6.5 + 11.2i)T^{2} \)
19 \( 1 + 1.31iT - 19T^{2} \)
23 \( 1 + (1.24 - 4.66i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-1.64 - 6.14i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (5.87 + 1.57i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-0.0561 + 0.0561i)T - 37iT^{2} \)
41 \( 1 + (0.875 - 3.26i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-0.590 - 0.340i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.445 - 0.771i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 3.82iT - 53T^{2} \)
59 \( 1 + (-10.3 + 5.96i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-9.62 + 2.57i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (-2.54 - 4.40i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.14 + 2.14i)T - 71iT^{2} \)
73 \( 1 + (-0.612 + 0.612i)T - 73iT^{2} \)
79 \( 1 + (0.617 - 0.165i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (8.24 + 4.75i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 16.7T + 89T^{2} \)
97 \( 1 + (-0.491 - 1.83i)T + (-84.0 + 48.5i)T^{2} \)
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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

−13.22907799489548379400016647617, −12.56494642574549097031077467120, −11.41083210751868343426735392565, −9.941785975471094860558164804658, −8.944993908745198453936374360054, −7.78885784587037783761826956840, −6.62085765277346879482340476180, −5.27850282287987249082649291181, −3.98418815477015698975247881258, −3.39017659736261628400284646273, 2.24115762356720819497095836727, 3.70780312996221897425661705284, 4.29151781699090271664331720864, 6.50567994131361513906614471550, 7.23120331527235411395167333391, 8.717616876411016923254925901081, 9.722841315383622557018899104959, 11.51459937867079214389486251289, 11.93452922515460636596708581410, 12.70390595647359557848286132208

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