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Properties

Label 2-153-153.106-c1-0-5
Degree $2$
Conductor $153$
Sign $0.899 + 0.437i$
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  + (−0.145 − 0.0840i)2-s + (−1.52 + 0.814i)3-s + (−0.985 − 1.70i)4-s + (1.69 + 0.454i)5-s + (0.290 + 0.00998i)6-s + (2.66 − 0.714i)7-s + 0.667i·8-s + (1.67 − 2.48i)9-s + (−0.208 − 0.208i)10-s + (3.67 − 0.983i)11-s + (2.89 + 1.80i)12-s + (−1.65 − 2.86i)13-s + (−0.448 − 0.120i)14-s + (−2.96 + 0.686i)15-s + (−1.91 + 3.31i)16-s + (4.06 − 0.716i)17-s + ⋯
L(s)  = 1  + (−0.102 − 0.0593i)2-s + (−0.882 + 0.469i)3-s + (−0.492 − 0.853i)4-s + (0.759 + 0.203i)5-s + (0.118 + 0.00407i)6-s + (1.00 − 0.270i)7-s + 0.235i·8-s + (0.558 − 0.829i)9-s + (−0.0660 − 0.0660i)10-s + (1.10 − 0.296i)11-s + (0.836 + 0.521i)12-s + (−0.458 − 0.794i)13-s + (−0.119 − 0.0320i)14-s + (−0.765 + 0.177i)15-s + (−0.478 + 0.829i)16-s + (0.984 − 0.173i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 + 0.437i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.899 + 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $0.899 + 0.437i$
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: \((2,\ 153,\ (\ :1/2),\ 0.899 + 0.437i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.906872 - 0.208849i\)
\(L(\frac12)\) \(\approx\) \(0.906872 - 0.208849i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.52 - 0.814i)T \)
17 \( 1 + (-4.06 + 0.716i)T \)
good2 \( 1 + (0.145 + 0.0840i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-1.69 - 0.454i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (-2.66 + 0.714i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-3.67 + 0.983i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (1.65 + 2.86i)T + (-6.5 + 11.2i)T^{2} \)
19 \( 1 + 0.246iT - 19T^{2} \)
23 \( 1 + (1.90 - 7.09i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (0.151 + 0.565i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (9.59 + 2.57i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-3.78 + 3.78i)T - 37iT^{2} \)
41 \( 1 + (-1.16 + 4.34i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-8.58 - 4.95i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.06 - 8.77i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 2.31iT - 53T^{2} \)
59 \( 1 + (4.27 - 2.46i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.40 + 1.44i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (-2.97 - 5.14i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.52 - 1.52i)T - 71iT^{2} \)
73 \( 1 + (4.27 - 4.27i)T - 73iT^{2} \)
79 \( 1 + (6.95 - 1.86i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (1.05 + 0.607i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 6.20T + 89T^{2} \)
97 \( 1 + (3.69 + 13.7i)T + (-84.0 + 48.5i)T^{2} \)
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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

−12.89254552995888730448700181283, −11.56105338394836610546210240324, −10.88035252796041107247196837571, −9.849352079297893388641213574835, −9.292872913972527200101492119387, −7.57298578721727402947528284557, −5.93649548245388320381311382188, −5.43585406270720651986849853045, −4.09792568596021824757768278756, −1.34675036307496568036027344940, 1.80921214705281296721713478982, 4.26550891623543887521666398079, 5.34231673353772395533735330815, 6.67140669891874683019562944912, 7.75977915268174132949410555420, 8.893735316346519623333877794383, 9.933127365982031406850890614420, 11.35935573014405933165159780528, 12.12400407017393894612587047466, 12.78680046397716233993020029581

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